Theorem ondif1 7221

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 7220	. 2 |- ( A e. ( On \ 1o ) <-> ( A e. On /\ A =/= (/) ) )
2		on0eln0 5485	. . 3 |- ( A e. On -> ( (/) e. A <-> A =/= (/) ) )
3	2	pm5.32i 649	. 2 |- ( ( A e. On /\ (/) e. A ) <-> ( A e. On /\ A =/= (/) ) )
4	1, 3	bitr4i 260	1 |- ( A e. ( On \ 1o ) <-> ( A e. On /\ (/) e. A ) )

Syntax hints:   <-> wb 189   /\ wa 376   e. wcel 1904   =/= wne 2641   \ cdif 3387   (/)c0 3722   Oncon0 5430   1oc1o 7193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-tr 4491  df-eprel 4750  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-ord 5433  df-on 5434  df-suc 5436  df-1o 7200

This theorem is referenced by:  cantnflem2  8213  oef1o  8221  cnfcom3  8227  infxpenc  8467