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Theorem ordsseleq 5790
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 5789 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴𝐵))
21orbi1d 739 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴𝐵𝐴 = 𝐵) ↔ (𝐴𝐵𝐴 = 𝐵)))
3 sspss 3739 . 2 (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
42, 3syl6rbbr 279 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  wss 3607  wpss 3608  Ord word 5760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764
This theorem is referenced by:  ordtri3or  5793  ordtri1  5794  ordtri2  5796  onsseleq  5803  ordsssuc  5850  ordsson  7031  ordsucelsuc  7064  limom  7122  onfununi  7483  cfslbn  9127  noextenddif  31946  finxpsuclem  33364
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