Theorem elvv 5730

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 5678	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
2		vex 3454	. . . . 5 ⊢ 𝑥 ∈ V
3		vex 3454	. . . . 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 2145  Vcvv 3450  ⟨cop 4590   × cxp 5653

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 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

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 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-un 3904  df-in 3906  df-ss 3916  df-sn 4585  df-pr 4587  df-op 4591  df-opab 5168  df-xp 5661

This theorem is used by:  elvvv  5731  elvvuni  5732  elrel  5778  copsex2gb  5787  relop  5830  elreldm  5919  dmsnn0  6203  funsndifnop  7148  1stval2  8003  2ndval2  8004  1st2val  8014  2nd2val  8015  dfopab2  8049  dfoprab3s  8050  dftpos4  8243  tpostpos  8244  fundmen  9038  cnvfi  9170  fundmge2nop0  14567  ssrelf  33088  fineqvac  35642  dfdm5  36352  dfrn5  36353  brtxp2  36458  pprodss4v  36461  brpprod3a  36463  brimg  36514  brxrn2  39132  fun2dmnopgexmpl  48172