Theorem ordelssne 5750

Description: For ordinal classes, membership is equivalent to strict inclusion. Corollary 7.8 of [TakeutiZaring] p. 37. (Contributed by NM, 25-Nov-1995.)

Assertion

Ref	Expression
ordelssne	|- ( ( Ord A /\ Ord B ) -> ( A e. B <-> ( A C_ B /\ A =/= B ) ) )

Proof of Theorem ordelssne

Step	Hyp	Ref	Expression
1		ordtr 5737	. . 3 |- ( Ord A -> Tr A )
2		tz7.7 5749	. . 3 |- ( ( Ord B /\ Tr A ) -> ( A e. B <-> ( A C_ B /\ A =/= B ) ) )
3	1, 2	sylan2 491	. 2 |- ( ( Ord B /\ Ord A ) -> ( A e. B <-> ( A C_ B /\ A =/= B ) ) )
4	3	ancoms 469	1 |- ( ( Ord A /\ Ord B ) -> ( A e. B <-> ( A C_ B /\ A =/= B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   =/= wne 2794   C_ wss 3574   Tr wtr 4752   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-nul 3916  df-if 4087  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:  ordelpss  5751  onelpss  5764  orduniorsuc  7030  ominf  8172