Theorem elvv 4812

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 4766	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
2		vex 2816	. . . . 5 ⊢ 𝑥 ∈ V
3		vex 2816	. . . . 5 ⊢ 𝑦 ∈ V
4	2, 3	pm3.2i 272	. . . 4 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
5	4	biantru 302	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
6	5	2exbii 1655	. 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
7	1, 6	bitr4i 187	1 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2203  Vcvv 2813  ⟨cop 3692   × cxp 4747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-opab 4172  df-xp 4755

This theorem is referenced by:  elvvv  4813  elvvuni  4814  ssrel  4838  elrel  4852  relop  4905  elreldm  4983  dmsnm  5228  1stval2  6349  2ndval2  6350  dfopab2  6383  dfoprab3s  6384  dftpos4  6494  tpostpos  6495  fundmen  7047  fundm2domnop0  11220