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Theorem ordsseleq 5965
Description: For ordinal classes, inclusion is equivalent to membership or equality. (Contributed by NM, 25-Nov-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ordsseleq ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))

Proof of Theorem ordsseleq
StepHypRef Expression
1 ordelpss 5964 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴𝐵))
21orbi1d 931 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴𝐵𝐴 = 𝐵) ↔ (𝐴𝐵𝐴 = 𝐵)))
3 sspss 3904 . 2 (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
42, 3syl6rbbr 281 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wo 865   = wceq 1637  wcel 2156  wss 3769  wpss 3770  Ord word 5935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-tr 4947  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-ord 5939
This theorem is referenced by:  ordtri3or  5968  ordtri1  5969  ordtri2  5971  onsseleq  5977  ordsssuc  6023  ordsson  7215  ordsucelsuc  7248  limom  7306  onfununi  7670  cfslbn  9370  noextenddif  32140  finxpsuclem  33548
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