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

Theorem cnvssOLD 5217
 Description: Obsolete proof of cnvss 5216 as of 27-Apr-2021. Subset theorem for converse. (Contributed by NM, 22-Mar-1998.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
cnvssOLD (𝐴𝐵𝐴𝐵)

Proof of Theorem cnvssOLD
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3562 . . . 4 (𝐴𝐵 → (⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ 𝐵))
2 df-br 4584 . . . 4 (𝑦𝐴𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
3 df-br 4584 . . . 4 (𝑦𝐵𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
41, 2, 33imtr4g 284 . . 3 (𝐴𝐵 → (𝑦𝐴𝑥𝑦𝐵𝑥))
54ssopab2dv 4929 . 2 (𝐴𝐵 → {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
6 df-cnv 5046 . 2 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
7 df-cnv 5046 . 2 𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}
85, 6, 73sstr4g 3609 1 (𝐴𝐵𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977   ⊆ wss 3540  ⟨cop 4131   class class class wbr 4583  {copab 4642  ◡ccnv 5037 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-in 3547  df-ss 3554  df-br 4584  df-opab 4644  df-cnv 5046 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator