MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  soeq2 Structured version   Visualization version   GIF version

Theorem soeq2 5612
Description: Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
Assertion
Ref Expression
soeq2 (𝐴 = 𝐵 → (𝑅 Or 𝐴𝑅 Or 𝐵))

Proof of Theorem soeq2
StepHypRef Expression
1 soss 5610 . . . 4 (𝐴𝐵 → (𝑅 Or 𝐵𝑅 Or 𝐴))
2 soss 5610 . . . 4 (𝐵𝐴 → (𝑅 Or 𝐴𝑅 Or 𝐵))
31, 2anim12i 611 . . 3 ((𝐴𝐵𝐵𝐴) → ((𝑅 Or 𝐵𝑅 Or 𝐴) ∧ (𝑅 Or 𝐴𝑅 Or 𝐵)))
4 eqss 3992 . . 3 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 dfbi2 473 . . 3 ((𝑅 Or 𝐵𝑅 Or 𝐴) ↔ ((𝑅 Or 𝐵𝑅 Or 𝐴) ∧ (𝑅 Or 𝐴𝑅 Or 𝐵)))
63, 4, 53imtr4i 291 . 2 (𝐴 = 𝐵 → (𝑅 Or 𝐵𝑅 Or 𝐴))
76bicomd 222 1 (𝐴 = 𝐵 → (𝑅 Or 𝐴𝑅 Or 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wss 3944   Or wor 5589
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-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-cleq 2717  df-ral 3051  df-ss 3961  df-po 5590  df-so 5591
This theorem is referenced by:  weeq2  5667  wemapso2  9578  oemapso  9707  fin2i  10320  isfin2-2  10344  fin1a2lem10  10434  zorn2lem7  10527  zornn0g  10530  opsrtoslem2  22022  sltsolem1  27654  soeq12d  42604  aomclem1  42620
  Copyright terms: Public domain W3C validator