Theorem elvv 5737

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 5685	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
2		vex 3467	. . . . 5 ⊢ 𝑥 ∈ V
3		vex 3467	. . . . 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 1876	. 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 1567  ∃wex 1806   ∈ wcel 2149  Vcvv 3463  ⟨cop 4600   × cxp 5660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-un 3918  df-in 3920  df-ss 3930  df-sn 4595  df-pr 4597  df-op 4601  df-opab 5178  df-xp 5668

This theorem is referenced by:  elvvv  5738  elvvuni  5739  elrel  5785  copsex2gb  5794  relop  5837  elreldm  5926  dmsnn0  6209  funsndifnop  7149  1stval2  8002  2ndval2  8003  1st2val  8013  2nd2val  8014  dfopab2  8048  dfoprab3s  8049  dftpos4  8240  tpostpos  8241  fundmen  9027  cnvfi  9159  fundmge2nop0  14538  ssrelf  32900  fineqvac  35451  dfdm5  36163  dfrn5  36164  brtxp2  36269  pprodss4v  36272  brpprod3a  36274  brimg  36325  brxrn2  38922  fun2dmnopgexmpl  47909