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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.) Avoid ax-sep 5292, ax-nul 5299, ax-pr 5420. (Revised by SN, 11-Dec-2024.) |
Ref | Expression |
---|---|
elopaelxp | ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} → 𝐴 ∈ (V × V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3477 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | vex 3477 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | pm3.2i 471 | . . . . 5 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜓 → (𝑥 ∈ V ∧ 𝑦 ∈ V)) |
5 | 4 | ssopab2i 5543 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜓} ⊆ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} |
6 | df-xp 5675 | . . 3 ⊢ (V × V) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} | |
7 | 5, 6 | sseqtrri 4015 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜓} ⊆ (V × V) |
8 | 7 | sseli 3974 | 1 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} → 𝐴 ∈ (V × V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 Vcvv 3473 {copab 5203 × cxp 5667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3475 df-in 3951 df-ss 3961 df-opab 5204 df-xp 5675 |
This theorem is referenced by: bropaex12 5759 clwlkcompim 28902 linedegen 34943 opelopab3 36388 |
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