Theorem inf3lem5 4597

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 4600 for detailed description.

Hypotheses

Ref	Expression
inf3lem.1	|- G = {<.y, z>. | z = {w e. x | (w i^i x) (_ y}}
inf3lem.2	|- F = (rec(G, (/)) |` om)
inf3lem.3	|- A e. V
inf3lem.4	|- B e. V

Assertion

Ref	Expression
inf3lem5	|- ((x =/= (/) /\ x (_ U.x) -> ((A e. om /\ B e. A) -> (F` B) (. (F` A)))

Distinct variable group:   x,y,z,w

Proof of Theorem inf3lem5

Step	Hyp	Ref	Expression
1		elnn 3137	. . . 4 |- ((B e. A /\ A e. om) -> B e. om)
2	1	ancoms 436	. . 3 |- ((A e. om /\ B e. A) -> B e. om)
3		nnord 3135	. . . . . . . . 9 |- (A e. om -> Ord A)
4		ordsucss 3064	. . . . . . . . 9 |- (Ord A -> (B e. A -> suc B (_ A))
5	3, 4	syl 10	. . . . . . . 8 |- (A e. om -> (B e. A -> suc B (_ A))
6	5	adantr 389	. . . . . . 7 |- ((A e. om /\ B e. om) -> (B e. A -> suc B (_ A))
7		fveq2 3715	. . . . . . . . . . . 12 |- (v = suc B -> (F` v) = (F` suc B))
8	7	psseq2d 2137	. . . . . . . . . . 11 |- (v = suc B -> ((F` B) (. (F` v) <-> (F` B) (. (F` suc B)))
9	8	imbi2d 611	. . . . . . . . . 10 |- (v = suc B -> (((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` v)) <-> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` suc B))))
10		fveq2 3715	. . . . . . . . . . . 12 |- (v = u -> (F` v) = (F` u))
11	10	psseq2d 2137	. . . . . . . . . . 11 |- (v = u -> ((F` B) (. (F` v) <-> (F` B) (. (F` u)))
12	11	imbi2d 611	. . . . . . . . . 10 |- (v = u -> (((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` v)) <-> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` u))))
13		fveq2 3715	. . . . . . . . . . . 12 |- (v = suc u -> (F` v) = (F` suc u))
14	13	psseq2d 2137	. . . . . . . . . . 11 |- (v = suc u -> ((F` B) (. (F` v) <-> (F` B) (. (F` suc u)))
15	14	imbi2d 611	. . . . . . . . . 10 |- (v = suc u -> (((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` v)) <-> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` suc u))))
16		fveq2 3715	. . . . . . . . . . . 12 |- (v = A -> (F` v) = (F` A))
17	16	psseq2d 2137	. . . . . . . . . . 11 |- (v = A -> ((F` B) (. (F` v) <-> (F` B) (. (F` A)))
18	17	imbi2d 611	. . . . . . . . . 10 |- (v = A -> (((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` v)) <-> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` A))))
19		peano2b 3142	. . . . . . . . . . 11 |- (B e. om <-> suc B e. om)
20		inf3lem.1	. . . . . . . . . . . . 13 |- G = {<.y, z>. | z = {w e. x | (w i^i x) (_ y}}
21		inf3lem.2	. . . . . . . . . . . . 13 |- F = (rec(G, (/)) |` om)
22		inf3lem.4	. . . . . . . . . . . . 13 |- B e. V
23	20, 21, 22, 22	inf3lem4 4596	. . . . . . . . . . . 12 |- ((x =/= (/) /\ x (_ U.x) -> (B e. om -> (F` B) (. (F` suc B)))
24	23	com12 11	. . . . . . . . . . 11 |- (B e. om -> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` suc B)))
25	19, 24	sylbir 201	. . . . . . . . . 10 |- (suc B e. om -> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` suc B)))
26		visset 1809	. . . . . . . . . . . . . 14 |- u e. V
27	20, 21, 26, 22	inf3lem4 4596	. . . . . . . . . . . . 13 |- ((x =/= (/) /\ x (_ U.x) -> (u e. om -> (F` u) (. (F` suc u)))
28		psstr 2146	. . . . . . . . . . . . . 14 |- (((F` B) (. (F` u) /\ (F` u) (. (F` suc u)) -> (F` B) (. (F` suc u))
29	28	expcom 374	. . . . . . . . . . . . 13 |- ((F` u) (. (F` suc u) -> ((F` B) (. (F` u) -> (F` B) (. (F` suc u)))
30	27, 29	syl6com 53	. . . . . . . . . . . 12 |- (u e. om -> ((x =/= (/) /\ x (_ U.x) -> ((F` B) (. (F` u) -> (F` B) (. (F` suc u))))
31	30	a2d 13	. . . . . . . . . . 11 |- (u e. om -> (((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` u)) -> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` suc u))))
32	31	ad2antrr 404	. . . . . . . . . 10 |- (((u e. om /\ suc B e. om) /\ suc B (_ u) -> (((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` u)) -> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` suc u))))
33	9, 12, 15, 18, 25, 32	findsg 3152	. . . . . . . . 9 |- (((A e. om /\ suc B e. om) /\ suc B (_ A) -> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` A)))
34	33	ex 373	. . . . . . . 8 |- ((A e. om /\ suc B e. om) -> (suc B (_ A -> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` A))))
35	34, 19	sylan2b 452	. . . . . . 7 |- ((A e. om /\ B e. om) -> (suc B (_ A -> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` A))))
36	6, 35	syld 27	. . . . . 6 |- ((A e. om /\ B e. om) -> (B e. A -> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` A))))
37	36	ex 373	. . . . 5 |- (A e. om -> (B e. om -> (B e. A -> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` A)))))
38	37	com23 32	. . . 4 |- (A e. om -> (B e. A -> (B e. om -> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` A)))))
39	38	imp 350	. . 3 |- ((A e. om /\ B e. A) -> (B e. om -> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` A))))
40	2, 39	mpd 26	. 2 |- ((A e. om /\ B e. A) -> ((x =/= (/) /\ x (_ U.x) -> (F` B) (. (F` A)))
41	40	com12 11	1 |- ((x =/= (/) /\ x (_ U.x) -> ((A e. om /\ B e. A) -> (F` B) (. (F` A)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 954   e. wcel 956   =/= wne 1582  {crab 1645  Vcvv 1807   i^i cin 2042   (_ wss 2043   (. wpss 2044  (/)c0 2276  U.cuni 2498  {copab 2661  Ord word 2942  suc csuc 2945  omcom 3126   |` cres 3167  ` cfv 3177  reccrdg 3922

This theorem is referenced by:  inf3lem6 4598

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-reg 4573

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193  df-rdg 3923

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