Theorem ondif2 7582

Description: Two ways to say that A is an ordinal greater than one. (Contributed by Mario Carneiro, 21-May-2015.)

Assertion

Ref	Expression
ondif2	|- ( A e. ( On \ 2o ) <-> ( A e. On /\ 1o e. A ) )

Proof of Theorem ondif2

Step	Hyp	Ref	Expression
1		eldif 3584	. 2 |- ( A e. ( On \ 2o ) <-> ( A e. On /\ -. A e. 2o ) )
2		1on 7567	. . . . 5 |- 1o e. On
3		ontri1 5757	. . . . . 6 |- ( ( A e. On /\ 1o e. On ) -> ( A C_ 1o <-> -. 1o e. A ) )
4		onsssuc 5813	. . . . . . 7 |- ( ( A e. On /\ 1o e. On ) -> ( A C_ 1o <-> A e. suc 1o ) )
5		df-2o 7561	. . . . . . . 8 |- 2o = suc 1o
6	5	eleq2i 2693	. . . . . . 7 |- ( A e. 2o <-> A e. suc 1o )
7	4, 6	syl6bbr 278	. . . . . 6 |- ( ( A e. On /\ 1o e. On ) -> ( A C_ 1o <-> A e. 2o ) )
8	3, 7	bitr3d 270	. . . . 5 |- ( ( A e. On /\ 1o e. On ) -> ( -. 1o e. A <-> A e. 2o ) )
9	2, 8	mpan2 707	. . . 4 |- ( A e. On -> ( -. 1o e. A <-> A e. 2o ) )
10	9	con1bid 345	. . 3 |- ( A e. On -> ( -. A e. 2o <-> 1o e. A ) )
11	10	pm5.32i 669	. 2 |- ( ( A e. On /\ -. A e. 2o ) <-> ( A e. On /\ 1o e. A ) )
12	1, 11	bitri 264	1 |- ( A e. ( On \ 2o ) <-> ( A e. On /\ 1o e. A ) )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   e. wcel 1990   \ cdif 3571   C_ wss 3574   Oncon0 5723   suc csuc 5725   1oc1o 7553   2oc2o 7554

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-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-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-suc 5729  df-1o 7560  df-2o 7561

This theorem is referenced by:  dif20el  7585  oeordi  7667  oewordi  7671  oaabs2  7725  omabs  7727  cnfcom3clem  8602  infxpenc2lem1  8842