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Mirrors > Home > MPE Home > Th. List > elvv | Structured version Visualization version GIF version |
Description: Membership in universal class of ordered pairs. (Contributed by NM, 4-Jul-1994.) |
Ref | Expression |
---|---|
elvv | ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp 5661 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))) | |
2 | vex 3452 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | vex 3452 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | pm3.2i 472 | . . . 4 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) |
5 | 4 | biantru 531 | . . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))) |
6 | 5 | 2exbii 1852 | . 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))) |
7 | 1, 6 | bitr4i 278 | 1 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 Vcvv 3448 ⟨cop 4597 × cxp 5636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-opab 5173 df-xp 5644 |
This theorem is referenced by: elvvv 5712 elvvuni 5713 elopaelxpOLD 5727 elrel 5759 copsex2gb 5767 relop 5811 elreldm 5895 dmsnn0 6164 funsndifnop 7102 1stval2 7943 2ndval2 7944 1st2val 7954 2nd2val 7955 dfopab2 7989 dfoprab3s 7990 dftpos4 8181 tpostpos 8182 fundmen 8982 cnvfi 9131 fundmge2nop0 14398 ssrelf 31576 fineqvac 33738 dfdm5 34386 dfrn5 34387 brtxp2 34495 pprodss4v 34498 brpprod3a 34500 brimg 34551 brxrn2 36866 fun2dmnopgexmpl 45590 |
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