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Theorem ordelpss 6383
Description: For ordinal classes, membership is equivalent to strict inclusion. Corollary 7.8 of [TakeutiZaring] p. 37. (Contributed by NM, 17-Jun-1998.)
Assertion
Ref Expression
ordelpss ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴𝐵))

Proof of Theorem ordelpss
StepHypRef Expression
1 ordelssne 6382 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))
2 df-pss 3960 . 2 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
31, 2bitr4di 289 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wcel 2098  wne 2932  wss 3941  wpss 3942  Ord word 6354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-tr 5257  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-ord 6358
This theorem is referenced by:  ordsseleq  6384  ordtri3or  6387  ordtr2  6399  onpsssuc  7801  php4  9210  nndomog  9213  nndomogOLD  9223  onomeneq  9225  findcard3  9282  ordsssucb  42634  oaun3lem1  42673  oaun3lem2  42674  ordpss  43759
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