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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 5296, ax-nul 5306, ax-pr 5432. (Revised by SN, 11-Dec-2024.) | 
| Ref | Expression | 
|---|---|
| elopaelxp | ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} → 𝐴 ∈ (V × V)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | vex 3484 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 2 | vex 3484 | . . . . . 6 ⊢ 𝑦 ∈ V | |
| 3 | 1, 2 | pm3.2i 470 | . . . . 5 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) | 
| 4 | 3 | a1i 11 | . . . 4 ⊢ (𝜓 → (𝑥 ∈ V ∧ 𝑦 ∈ V)) | 
| 5 | 4 | ssopab2i 5555 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜓} ⊆ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} | 
| 6 | df-xp 5691 | . . 3 ⊢ (V × V) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} | |
| 7 | 5, 6 | sseqtrri 4033 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜓} ⊆ (V × V) | 
| 8 | 7 | sseli 3979 | 1 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜓} → 𝐴 ∈ (V × V)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 {copab 5205 × cxp 5683 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-opab 5206 df-xp 5691 | 
| This theorem is referenced by: bropaex12 5777 clwlkcompim 29800 linedegen 36144 opelopab3 37725 | 
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