Theorem phplem2 4495

Description: Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus one of its elements.

Hypotheses

Ref	Expression
phplem2.1	|- A e. V
phplem2.2	|- B e. V

Assertion

Ref	Expression
phplem2	|- ((A e. om /\ B e. A) -> A ~~ (suc A \ {B}))

Proof of Theorem phplem2

Step	Hyp	Ref	Expression
1		difss 2163	. . . . . . . . . 10 |- (A \ {B}) (_ A
2		ssrin 2230	. . . . . . . . . 10 |- ((A \ {B}) (_ A -> ((A \ {B}) i^i {A}) (_ (A i^i {A}))
3	1, 2	ax-mp 7	. . . . . . . . 9 |- ((A \ {B}) i^i {A}) (_ (A i^i {A})
4		nnord 3135	. . . . . . . . . . 11 |- (A e. om -> Ord A)
5		orddisj 2980	. . . . . . . . . . 11 |- (Ord A -> (A i^i {A}) = (/))
6	4, 5	syl 10	. . . . . . . . . 10 |- (A e. om -> (A i^i {A}) = (/))
7	6	sseq2d 2085	. . . . . . . . 9 |- (A e. om -> (((A \ {B}) i^i {A}) (_ (A i^i {A}) <-> ((A \ {B}) i^i {A}) (_ (/)))
8	3, 7	mpbii 193	. . . . . . . 8 |- (A e. om -> ((A \ {B}) i^i {A}) (_ (/))
9		ss0 2299	. . . . . . . 8 |- (((A \ {B}) i^i {A}) (_ (/) -> ((A \ {B}) i^i {A}) = (/))
10	8, 9	syl 10	. . . . . . 7 |- (A e. om -> ((A \ {B}) i^i {A}) = (/))
11		incom 2204	. . . . . . 7 |- ({A} i^i (A \ {B})) = ((A \ {B}) i^i {A})
12	10, 11	syl5eq 1516	. . . . . 6 |- (A e. om -> ({A} i^i (A \ {B})) = (/))
13		difdisj 2333	. . . . . 6 |- ({B} i^i (A \ {B})) = (/)
14	12, 13	jctil 292	. . . . 5 |- (A e. om -> (({B} i^i (A \ {B})) = (/) /\ ({A} i^i (A \ {B})) = (/)))
15		phplem2.2	. . . . . . . 8 |- B e. V
16		phplem2.1	. . . . . . . 8 |- A e. V
17	15, 16	f1osn 3710	. . . . . . 7 |- {<.B, A>.}:{B}-1-1-onto->{A}
18		snex 2745	. . . . . . . 8 |- {B} e. V
19	18	f1oen 4385	. . . . . . 7 |- ({<.B, A>.}:{B}-1-1-onto->{A} -> {B} ~~ {A})
20	17, 19	ax-mp 7	. . . . . 6 |- {B} ~~ {A}
21	16, 1	ssexi 2715	. . . . . . 7 |- (A \ {B}) e. V
22	21	enref 4378	. . . . . 6 |- (A \ {B}) ~~ (A \ {B})
23	20, 22	pm3.2i 285	. . . . 5 |- ({B} ~~ {A} /\ (A \ {B}) ~~ (A \ {B}))
24	14, 23	jctil 292	. . . 4 |- (A e. om -> (({B} ~~ {A} /\ (A \ {B}) ~~ (A \ {B})) /\ (({B} i^i (A \ {B})) = (/) /\ ({A} i^i (A \ {B})) = (/))))
25		unen 4420	. . . 4 |- ((({B} ~~ {A} /\ (A \ {B}) ~~ (A \ {B})) /\ (({B} i^i (A \ {B})) = (/) /\ ({A} i^i (A \ {B})) = (/))) -> ({B} u. (A \ {B})) ~~ ({A} u. (A \ {B})))
26	24, 25	syl 10	. . 3 |- (A e. om -> ({B} u. (A \ {B})) ~~ ({A} u. (A \ {B})))
27	26	adantr 389	. 2 |- ((A e. om /\ B e. A) -> ({B} u. (A \ {B})) ~~ ({A} u. (A \ {B})))
28		difsnid 2463	. . . 4 |- (B e. A -> ((A \ {B}) u. {B}) = A)
29		uncom 2172	. . . 4 |- ({B} u. (A \ {B})) = ((A \ {B}) u. {B})
30	28, 29	syl5eq 1516	. . 3 |- (B e. A -> ({B} u. (A \ {B})) = A)
31	30	adantl 388	. 2 |- ((A e. om /\ B e. A) -> ({B} u. (A \ {B})) = A)
32		phplem1 4494	. 2 |- ((A e. om /\ B e. A) -> ({A} u. (A \ {B})) = (suc A \ {B}))
33	27, 31, 32	3brtr3d 2639	1 |- ((A e. om /\ B e. A) -> A ~~ (suc A \ {B}))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 954   e. wcel 956  Vcvv 1807   \ cdif 2040   u. cun 2041   i^i cin 2042   (_ wss 2043  (/)c0 2276  {csn 2405  <.cop 2407   class class class wbr 2614  Ord word 2942  suc csuc 2945  omcom 3126  -1-1-onto->wf1o 3176   ~~ cen 4354

This theorem is referenced by:  phplem3 4496

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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  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-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-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-en 4357

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