Theorem ord0eln0 5779

Description: A nonempty ordinal contains the empty set. (Contributed by NM, 25-Nov-1995.)

Assertion

Ref	Expression
ord0eln0	|- ( Ord A -> ( (/) e. A <-> A =/= (/) ) )

Proof of Theorem ord0eln0

Step	Hyp	Ref	Expression
1		ne0i 3921	. 2 |- ( (/) e. A -> A =/= (/) )
2		ord0 5777	. . . 4 |- Ord (/)
3		noel 3919	. . . . 5 |- -. A e. (/)
4		ordtri2 5758	. . . . . 6 |- ( ( Ord A /\ Ord (/) ) -> ( A e. (/) <-> -. ( A = (/) \/ (/) e. A ) ) )
5	4	con2bid 344	. . . . 5 |- ( ( Ord A /\ Ord (/) ) -> ( ( A = (/) \/ (/) e. A ) <-> -. A e. (/) ) )
6	3, 5	mpbiri 248	. . . 4 |- ( ( Ord A /\ Ord (/) ) -> ( A = (/) \/ (/) e. A ) )
7	2, 6	mpan2 707	. . 3 |- ( Ord A -> ( A = (/) \/ (/) e. A ) )
8		neor 2885	. . 3 |- ( ( A = (/) \/ (/) e. A ) <-> ( A =/= (/) -> (/) e. A ) )
9	7, 8	sylib 208	. 2 |- ( Ord A -> ( A =/= (/) -> (/) e. A ) )
10	1, 9	impbid2 216	1 |- ( Ord A -> ( (/) e. A <-> A =/= (/) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   Ord word 5722

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

This theorem is referenced by:  on0eln0  5780  dflim2  5781  0ellim  5787  0elsuc  7035  ordge1n0  7578  omwordi  7651  omass  7660  nnmord  7712  nnmwordi  7715  wemapwe  8594  elni2  9699  bnj529  30811