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Theorem ordelssne 6380
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 6367 . . 3 (Ord 𝐴 → Tr 𝐴)
2 tz7.7 6379 . . 3 ((Ord 𝐵 ∧ Tr 𝐴) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))
31, 2sylan2 593 . 2 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))
43ancoms 459 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106  wne 2939  wss 3944  Tr wtr 5258  Ord word 6352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6356
This theorem is referenced by:  ordelpss  6381  onelpss  6393  orduniorsuc  7801  ominf  9241  ominfOLD  9242  scutbdaybnd2lim  27244
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