Theorem ondif1 7106

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 7105	. 2 |- ( A e. ( On \ 1o ) <-> ( A e. On /\ A =/= (/) ) )
2		on0eln0 4874	. . 3 |- ( A e. On -> ( (/) e. A <-> A =/= (/) ) )
3	2	pm5.32i 635	. 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 367   e. wcel 1840   =/= wne 2596   \ cdif 3408   (/)c0 3735   Oncon0 4819   1oc1o 7078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pr 4627

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-tr 4487  df-eprel 4731  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-suc 4825  df-1o 7085

This theorem is referenced by:  cantnflem2  8059  oef1o  8091  oef1oOLD  8092  cnfcom3  8098  cnfcom3OLD  8106  infxpenc  8345  infxpencOLD  8350