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Mirrors > Home > MPE Home > Th. List > elopaelxp | Structured version Visualization version GIF version |
Description: Membership in an ordered-pair class abstraction implies membership in a Cartesian product. (Contributed by Alexander van der Vekens, 23-Jun-2018.) |
Ref | Expression |
---|---|
elopaelxp | ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} → 𝐴 ∈ (V × V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 486 | . . 3 ⊢ ((𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜓) → 𝐴 = 〈𝑥, 𝑦〉) | |
2 | 1 | 2eximi 1843 | . 2 ⊢ (∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜓) → ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) |
3 | elopab 5393 | . 2 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜓)) | |
4 | elvv 5608 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
5 | 2, 3, 4 | 3imtr4i 295 | 1 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} → 𝐴 ∈ (V × V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2112 Vcvv 3398 〈cop 4533 {copab 5101 × cxp 5534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-v 3400 df-dif 3856 df-un 3858 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-opab 5102 df-xp 5542 |
This theorem is referenced by: bropaex12 5624 clwlkcompim 27821 linedegen 34131 opelopab3 35561 |
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