Theorem ondif1 7581

Description: Two ways to say that A is a nonzero ordinal number. (Contributed by Mario Carneiro, 21-May-2015.)

Assertion

Ref	Expression
ondif1	|- ( A e. ( On \ 1o ) <-> ( A e. On /\ (/) e. A ) )

Proof of Theorem ondif1

Step	Hyp	Ref	Expression
1		dif1o 7580	. 2 |- ( A e. ( On \ 1o ) <-> ( A e. On /\ A =/= (/) ) )
2		on0eln0 5780	. . 3 |- ( A e. On -> ( (/) e. A <-> A =/= (/) ) )
3	2	pm5.32i 669	. 2 |- ( ( A e. On /\ (/) e. A ) <-> ( A e. On /\ A =/= (/) ) )
4	1, 3	bitr4i 267	1 |- ( A e. ( On \ 1o ) <-> ( A e. On /\ (/) e. A ) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   =/= wne 2794   \ cdif 3571   (/)c0 3915   Oncon0 5723   1oc1o 7553

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-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

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-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

This theorem is referenced by:  cantnflem2  8587  oef1o  8595  cnfcom3  8601  infxpenc  8841