MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  bropaex12 Structured version   Visualization version   GIF version

Theorem bropaex12 5101
Description: Two classes related by an ordered pair class builder are sets. (Contributed by AV, 21-Jan-2020.)
Hypothesis
Ref Expression
bropaex12.1 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓}
Assertion
Ref Expression
bropaex12 (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem bropaex12
StepHypRef Expression
1 df-br 4574 . . . 4 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2 bropaex12.1 . . . . 5 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓}
32eleq2i 2675 . . . 4 (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
41, 3bitri 262 . . 3 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
5 elopaelxp 5100 . . 3 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → ⟨𝐴, 𝐵⟩ ∈ (V × V))
64, 5sylbi 205 . 2 (𝐴𝑅𝐵 → ⟨𝐴, 𝐵⟩ ∈ (V × V))
7 opelxp 5056 . 2 (⟨𝐴, 𝐵⟩ ∈ (V × V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
86, 7sylib 206 1 (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  Vcvv 3168  cop 4126   class class class wbr 4573  {copab 4632   × cxp 5022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-br 4574  df-opab 4634  df-xp 5030
This theorem is referenced by:  fpwwe  9320  efgrelexlema  17927  clcllaw  41615  asslawass  41617
  Copyright terms: Public domain W3C validator