Theorem elvv 5736

Description: Membership in universal class of ordered pairs. (Contributed by NM, 4-Jul-1994.)

Assertion

Ref	Expression
elvv	⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem elvv

Step	Hyp	Ref	Expression
1		elxp 5684	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
2		vex 3459	. . . . 5 ⊢ 𝑥 ∈ V
3		vex 3459	. . . . 5 ⊢ 𝑦 ∈ V
4	2, 3	pm3.2i 475	. . . 4 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
5	4	biantru 538	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
6	5	2exbii 1879	. 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
7	1, 6	bitr4i 281	1 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  Vcvv 3455  ⟨cop 4595   × cxp 5659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-un 3910  df-in 3912  df-ss 3922  df-sn 4590  df-pr 4592  df-op 4596  df-opab 5174  df-xp 5667

This theorem is referenced by:  elvvv  5737  elvvuni  5738  elrel  5784  copsex2gb  5793  relop  5836  elreldm  5925  dmsnn0  6208  funsndifnop  7148  1stval2  7999  2ndval2  8000  1st2val  8010  2nd2val  8011  dfopab2  8045  dfoprab3s  8046  dftpos4  8237  tpostpos  8238  fundmen  9024  cnvfi  9156  fundmge2nop0  14535  ssrelf  32960  fineqvac  35529  dfdm5  36265  dfrn5  36266  brtxp2  36371  pprodss4v  36374  brpprod3a  36376  brimg  36427  brxrn2  39033  fun2dmnopgexmpl  48021