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Theorem ordelssne 6410
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) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))

Proof of Theorem ordelssne
StepHypRef Expression
1 ordtr 6397 . . 3 (Ord 𝐴 → Tr 𝐴)
2 tz7.7 6409 . . 3 ((Ord 𝐵 ∧ Tr 𝐴) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))
31, 2sylan2 593 . 2 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))
43ancoms 458 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2107  wne 2939  wss 3950  Tr wtr 5258  Ord word 6382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386
This theorem is referenced by:  ordelpss  6411  onelpss  6423  orduniorsuc  7851  ominf  9295  ominfOLD  9296  scutbdaybnd2lim  27863
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