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Theorem ordelssne 6403
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 6390 . . 3 (Ord 𝐴 → Tr 𝐴)
2 tz7.7 6402 . . 3 ((Ord 𝐵 ∧ Tr 𝐴) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))
31, 2sylan2 591 . 2 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))
43ancoms 457 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wcel 2099  wne 2930  wss 3947  Tr wtr 5270  Ord word 6375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-tr 5271  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6379
This theorem is referenced by:  ordelpss  6404  onelpss  6416  orduniorsuc  7839  ominf  9292  ominfOLD  9293  scutbdaybnd2lim  27847
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