Theorem onomeneq 4498

Description: An ordinal number equinumerous to a natural number is equal to it. Proposition 10.22 of [TakeutiZaring] p. 90 and its converse.

Assertion

Ref	Expression
onomeneq	|- ((A e. On /\ B e. om) -> (A ~~ B <-> A = B))

Proof of Theorem onomeneq

Step	Hyp	Ref	Expression
1		nneneq 4492	. . . . 5 |- ((A e. om /\ B e. om) -> (A ~~ B <-> A = B))
2	1	biimpa 416	. . . 4 |- (((A e. om /\ B e. om) /\ A ~~ B) -> A = B)
3		php5 4497	. . . . . . . . . 10 |- (B e. om -> -. B ~~ suc B)
4	3	adantr 389	. . . . . . . . 9 |- ((B e. om /\ A ~~ B) -> -. B ~~ suc B)
5		enen1 4457	. . . . . . . . 9 |- ((B e. om /\ A ~~ B) -> (A ~~ suc B <-> B ~~ suc B))
6	4, 5	mtbird 713	. . . . . . . 8 |- ((B e. om /\ A ~~ B) -> -. A ~~ suc B)
7	6	adantll 392	. . . . . . 7 |- (((A e. On /\ B e. om) /\ A ~~ B) -> -. A ~~ suc B)
8		endomtr 4401	. . . . . . . . . . . . 13 |- ((A ~~ B /\ B ~<_ suc B) -> A ~<_ suc B)
9		sssucid 3037	. . . . . . . . . . . . . 14 |- B (_ suc B
10		ssdomg 4389	. . . . . . . . . . . . . 14 |- (B e. om -> (B (_ suc B -> B ~<_ suc B))
11	9, 10	mpi 44	. . . . . . . . . . . . 13 |- (B e. om -> B ~<_ suc B)
12	8, 11	sylan2 451	. . . . . . . . . . . 12 |- ((A ~~ B /\ B e. om) -> A ~<_ suc B)
13	12	ancoms 436	. . . . . . . . . . 11 |- ((B e. om /\ A ~~ B) -> A ~<_ suc B)
14	13	a1d 12	. . . . . . . . . 10 |- ((B e. om /\ A ~~ B) -> (om (_ A -> A ~<_ suc B))
15	14	adantll 392	. . . . . . . . 9 |- (((A e. On /\ B e. om) /\ A ~~ B) -> (om (_ A -> A ~<_ suc B))
16		ssel 2053	. . . . . . . . . . . . . . 15 |- (om (_ A -> (B e. om -> B e. A))
17	16	com12 11	. . . . . . . . . . . . . 14 |- (B e. om -> (om (_ A -> B e. A))
18	17	adantr 389	. . . . . . . . . . . . 13 |- ((B e. om /\ A e. On) -> (om (_ A -> B e. A))
19		ordelsuc 3061	. . . . . . . . . . . . . 14 |- ((B e. om /\ Ord A) -> (B e. A <-> suc B (_ A))
20		eloni 2948	. . . . . . . . . . . . . 14 |- (A e. On -> Ord A)
21	19, 20	sylan2 451	. . . . . . . . . . . . 13 |- ((B e. om /\ A e. On) -> (B e. A <-> suc B (_ A))
22	18, 21	sylibd 202	. . . . . . . . . . . 12 |- ((B e. om /\ A e. On) -> (om (_ A -> suc B (_ A))
23		ssdom2g 4390	. . . . . . . . . . . . 13 |- (A e. On -> (suc B (_ A -> suc B ~<_ A))
24	23	adantl 388	. . . . . . . . . . . 12 |- ((B e. om /\ A e. On) -> (suc B (_ A -> suc B ~<_ A))
25	22, 24	syld 27	. . . . . . . . . . 11 |- ((B e. om /\ A e. On) -> (om (_ A -> suc B ~<_ A))
26	25	ancoms 436	. . . . . . . . . 10 |- ((A e. On /\ B e. om) -> (om (_ A -> suc B ~<_ A))
27	26	adantr 389	. . . . . . . . 9 |- (((A e. On /\ B e. om) /\ A ~~ B) -> (om (_ A -> suc B ~<_ A))
28	15, 27	jcad 598	. . . . . . . 8 |- (((A e. On /\ B e. om) /\ A ~~ B) -> (om (_ A -> (A ~<_ suc B /\ suc B ~<_ A)))
29		sbth 4437	. . . . . . . 8 |- ((A ~<_ suc B /\ suc B ~<_ A) -> A ~~ suc B)
30	28, 29	syl6 22	. . . . . . 7 |- (((A e. On /\ B e. om) /\ A ~~ B) -> (om (_ A -> A ~~ suc B))
31	7, 30	mtod 108	. . . . . 6 |- (((A e. On /\ B e. om) /\ A ~~ B) -> -. om (_ A)
32		ordom 3131	. . . . . . . . . 10 |- Ord om
33		ordtri1 2970	. . . . . . . . . 10 |- ((Ord om /\ Ord A) -> (om (_ A <-> -. A e. om))
34	32, 33	mpan 693	. . . . . . . . 9 |- (Ord A -> (om (_ A <-> -. A e. om))
35	20, 34	syl 10	. . . . . . . 8 |- (A e. On -> (om (_ A <-> -. A e. om))
36	35	con2bid 524	. . . . . . 7 |- (A e. On -> (A e. om <-> -. om (_ A))
37	36	ad2antrr 404	. . . . . 6 |- (((A e. On /\ B e. om) /\ A ~~ B) -> (A e. om <-> -. om (_ A))
38	31, 37	mpbird 196	. . . . 5 |- (((A e. On /\ B e. om) /\ A ~~ B) -> A e. om)
39		simplr 413	. . . . 5 |- (((A e. On /\ B e. om) /\ A ~~ B) -> B e. om)
40	38, 39	jca 288	. . . 4 |- (((A e. On /\ B e. om) /\ A ~~ B) -> (A e. om /\ B e. om))
41		pm3.27 323	. . . 4 |- (((A e. On /\ B e. om) /\ A ~~ B) -> A ~~ B)
42	2, 40, 41	sylanc 471	. . 3 |- (((A e. On /\ B e. om) /\ A ~~ B) -> A = B)
43	42	ex 373	. 2 |- ((A e. On /\ B e. om) -> (A ~~ B -> A = B))
44		eqeng 4373	. . 3 |- (A e. On -> (A = B -> A ~~ B))
45	44	adantr 389	. 2 |- ((A e. On /\ B e. om) -> (A = B -> A ~~ B))
46	43, 45	impbid 514	1 |- ((A e. On /\ B e. om) -> (A ~~ B <-> A = B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955   (_ wss 2037   class class class wbr 2609  Ord word 2937  Oncon0 2938  suc csuc 2940  omcom 3121   ~~ cen 4348   ~<_ cdom 4349

This theorem is referenced by:  onfin 4499

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-er 4245  df-en 4351  df-dom 4352  df-sdom 4353

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