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Mirrors > Home > MPE Home > Th. List > elopaelxp | Structured version Visualization version GIF version |
Description: Membership in an ordered pair class builder 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 472 | . . 3 ⊢ ((𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜓) → 𝐴 = 〈𝑥, 𝑦〉) | |
2 | 1 | 2eximi 1753 | . 2 ⊢ (∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜓) → ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) |
3 | elopab 4908 | . 2 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜓)) | |
4 | elvv 5100 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
5 | 2, 3, 4 | 3imtr4i 280 | 1 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} → 𝐴 ∈ (V × V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 〈cop 4131 {copab 4642 × cxp 5036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-opab 4644 df-xp 5044 |
This theorem is referenced by: bropaex12 5115 wlkcompim 26054 clwlkcompim 26292 linedegen 31420 opelopab3 32681 clWlkcompim 40987 |
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