Theorem ondif1 7054

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 7053	. 2 |- ( A e. ( On \ 1o ) <-> ( A e. On /\ A =/= (/) ) )
2		on0eln0 4885	. . 3 |- ( A e. On -> ( (/) e. A <-> A =/= (/) ) )
3	2	pm5.32i 637	. 2 |- ( ( A e. On /\ (/) e. A ) <-> ( A e. On /\ A =/= (/) ) )
4	1, 3	bitr4i 252	1 |- ( A e. ( On \ 1o ) <-> ( A e. On /\ (/) e. A ) )

Syntax hints:   <-> wb 184   /\ wa 369   e. wcel 1758   =/= wne 2648   \ cdif 3436   (/)c0 3748   Oncon0 4830   1oc1o 7026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-tr 4497  df-eprel 4743  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-suc 4836  df-1o 7033

This theorem is referenced by:  cantnflem2  8012  oef1o  8044  oef1oOLD  8045  cnfcom3  8051  cnfcom3OLD  8059  infxpenc  8298  infxpencOLD  8303