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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 5261, ax-nul 5268, ax-pr 5389. (Revised by SN, 11-Dec-2024.) |
Ref | Expression |
---|---|
elopaelxp | ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → 𝐴 ∈ (V × V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3452 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | vex 3452 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | pm3.2i 472 | . . . . 5 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜓 → (𝑥 ∈ V ∧ 𝑦 ∈ V)) |
5 | 4 | ssopab2i 5512 | . . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} |
6 | df-xp 5644 | . . 3 ⊢ (V × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} | |
7 | 5, 6 | sseqtrri 3986 | . 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (V × V) |
8 | 7 | sseli 3945 | 1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → 𝐴 ∈ (V × V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 Vcvv 3448 {copab 5172 × 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 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3450 df-in 3922 df-ss 3932 df-opab 5173 df-xp 5644 |
This theorem is referenced by: bropaex12 5728 clwlkcompim 28770 linedegen 34757 opelopab3 36205 |
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