Theorem elvv 5738

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 5686	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
2		vex 3461	. . . . 5 ⊢ 𝑥 ∈ V
3		vex 3461	. . . . 5 ⊢ 𝑦 ∈ V
4	2, 3	pm3.2i 476	. . . 4 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
5	4	biantru 539	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
6	5	2exbii 1882	. 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⟩)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  Vcvv 3457  ⟨cop 4597   × cxp 5661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-un 3911  df-in 3913  df-ss 3923  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5176  df-xp 5669

This theorem is used by:  elvvv  5739  elvvuni  5740  elrel  5786  copsex2gb  5795  relop  5838  elreldm  5927  dmsnn0  6210  funsndifnop  7154  1stval2  8009  2ndval2  8010  1st2val  8020  2nd2val  8021  dfopab2  8055  dfoprab3s  8056  dftpos4  8247  tpostpos  8248  fundmen  9035  cnvfi  9167  fundmge2nop0  14557  ssrelf  33031  fineqvac  35586  dfdm5  36302  dfrn5  36303  brtxp2  36408  pprodss4v  36411  brpprod3a  36413  brimg  36464  brxrn2  39091  fun2dmnopgexmpl  48079