Theorem onomeneq 8150

Description: An ordinal number equinumerous to a natural number is equal to it. Proposition 10.22 of [TakeutiZaring] p. 90 and its converse. (Contributed by NM, 26-Jul-2004.)

Assertion

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

Proof of Theorem onomeneq

Step	Hyp	Ref	Expression
1		php5 8148	. . . . . . . . 9 |- ( B e. om -> -. B ~~ suc B )
2	1	ad2antlr 763	. . . . . . . 8 |- ( ( ( A e. On /\ B e. om ) /\ A ~~ B ) -> -. B ~~ suc B )
3		enen1 8100	. . . . . . . . 9 |- ( A ~~ B -> ( A ~~ suc B <-> B ~~ suc B ) )
4	3	adantl 482	. . . . . . . 8 |- ( ( ( A e. On /\ B e. om ) /\ A ~~ B ) -> ( A ~~ suc B <-> B ~~ suc B ) )
5	2, 4	mtbird 315	. . . . . . 7 |- ( ( ( A e. On /\ B e. om ) /\ A ~~ B ) -> -. A ~~ suc B )
6		peano2 7086	. . . . . . . . . . . . . 14 |- ( B e. om -> suc B e. om )
7		sssucid 5802	. . . . . . . . . . . . . 14 |- B C_ suc B
8		ssdomg 8001	. . . . . . . . . . . . . 14 |- ( suc B e. om -> ( B C_ suc B -> B ~<_ suc B ) )
9	6, 7, 8	mpisyl 21	. . . . . . . . . . . . 13 |- ( B e. om -> B ~<_ suc B )
10		endomtr 8014	. . . . . . . . . . . . 13 |- ( ( A ~~ B /\ B ~<_ suc B ) -> A ~<_ suc B )
11	9, 10	sylan2 491	. . . . . . . . . . . 12 |- ( ( A ~~ B /\ B e. om ) -> A ~<_ suc B )
12	11	ancoms 469	. . . . . . . . . . 11 |- ( ( B e. om /\ A ~~ B ) -> A ~<_ suc B )
13	12	a1d 25	. . . . . . . . . 10 |- ( ( B e. om /\ A ~~ B ) -> ( om C_ A -> A ~<_ suc B ) )
14	13	adantll 750	. . . . . . . . 9 |- ( ( ( A e. On /\ B e. om ) /\ A ~~ B ) -> ( om C_ A -> A ~<_ suc B ) )
15		ssel 3597	. . . . . . . . . . . . . . 15 |- ( om C_ A -> ( B e. om -> B e. A ) )
16	15	com12 32	. . . . . . . . . . . . . 14 |- ( B e. om -> ( om C_ A -> B e. A ) )
17	16	adantr 481	. . . . . . . . . . . . 13 |- ( ( B e. om /\ A e. On ) -> ( om C_ A -> B e. A ) )
18		eloni 5733	. . . . . . . . . . . . . 14 |- ( A e. On -> Ord A )
19		ordelsuc 7020	. . . . . . . . . . . . . 14 |- ( ( B e. om /\ Ord A ) -> ( B e. A <-> suc B C_ A ) )
20	18, 19	sylan2 491	. . . . . . . . . . . . 13 |- ( ( B e. om /\ A e. On ) -> ( B e. A <-> suc B C_ A ) )
21	17, 20	sylibd 229	. . . . . . . . . . . 12 |- ( ( B e. om /\ A e. On ) -> ( om C_ A -> suc B C_ A ) )
22		ssdomg 8001	. . . . . . . . . . . . 13 |- ( A e. On -> ( suc B C_ A -> suc B ~<_ A ) )
23	22	adantl 482	. . . . . . . . . . . 12 |- ( ( B e. om /\ A e. On ) -> ( suc B C_ A -> suc B ~<_ A ) )
24	21, 23	syld 47	. . . . . . . . . . 11 |- ( ( B e. om /\ A e. On ) -> ( om C_ A -> suc B ~<_ A ) )
25	24	ancoms 469	. . . . . . . . . 10 |- ( ( A e. On /\ B e. om ) -> ( om C_ A -> suc B ~<_ A ) )
26	25	adantr 481	. . . . . . . . 9 |- ( ( ( A e. On /\ B e. om ) /\ A ~~ B ) -> ( om C_ A -> suc B ~<_ A ) )
27	14, 26	jcad 555	. . . . . . . 8 |- ( ( ( A e. On /\ B e. om ) /\ A ~~ B ) -> ( om C_ A -> ( A ~<_ suc B /\ suc B ~<_ A ) ) )
28		sbth 8080	. . . . . . . 8 |- ( ( A ~<_ suc B /\ suc B ~<_ A ) -> A ~~ suc B )
29	27, 28	syl6 35	. . . . . . 7 |- ( ( ( A e. On /\ B e. om ) /\ A ~~ B ) -> ( om C_ A -> A ~~ suc B ) )
30	5, 29	mtod 189	. . . . . 6 |- ( ( ( A e. On /\ B e. om ) /\ A ~~ B ) -> -. om C_ A )
31		ordom 7074	. . . . . . . . 9 |- Ord om
32		ordtri1 5756	. . . . . . . . 9 |- ( ( Ord om /\ Ord A ) -> ( om C_ A <-> -. A e. om ) )
33	31, 18, 32	sylancr 695	. . . . . . . 8 |- ( A e. On -> ( om C_ A <-> -. A e. om ) )
34	33	con2bid 344	. . . . . . 7 |- ( A e. On -> ( A e. om <-> -. om C_ A ) )
35	34	ad2antrr 762	. . . . . 6 |- ( ( ( A e. On /\ B e. om ) /\ A ~~ B ) -> ( A e. om <-> -. om C_ A ) )
36	30, 35	mpbird 247	. . . . 5 |- ( ( ( A e. On /\ B e. om ) /\ A ~~ B ) -> A e. om )
37		simplr 792	. . . . 5 |- ( ( ( A e. On /\ B e. om ) /\ A ~~ B ) -> B e. om )
38	36, 37	jca 554	. . . 4 |- ( ( ( A e. On /\ B e. om ) /\ A ~~ B ) -> ( A e. om /\ B e. om ) )
39		nneneq 8143	. . . . 5 |- ( ( A e. om /\ B e. om ) -> ( A ~~ B <-> A = B ) )
40	39	biimpa 501	. . . 4 |- ( ( ( A e. om /\ B e. om ) /\ A ~~ B ) -> A = B )
41	38, 40	sylancom 701	. . 3 |- ( ( ( A e. On /\ B e. om ) /\ A ~~ B ) -> A = B )
42	41	ex 450	. 2 |- ( ( A e. On /\ B e. om ) -> ( A ~~ B -> A = B ) )
43		eqeng 7989	. . 3 |- ( A e. On -> ( A = B -> A ~~ B ) )
44	43	adantr 481	. 2 |- ( ( A e. On /\ B e. om ) -> ( A = B -> A ~~ B ) )
45	42, 44	impbid 202	1 |- ( ( A e. On /\ B e. om ) -> ( A ~~ B <-> A = B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   class class class wbr 4653   Ord word 5722   Oncon0 5723   suc csuc 5725   omcom 7065   ~~ cen 7952   ~<_ cdom 7953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958

This theorem is referenced by:  onfin  8151  ficardom  8787  finnisoeu  8936