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Mirrors > Home > MPE Home > Th. List > opelvvg | Structured version Visualization version GIF version |
Description: Ordered pair membership in the universal class of ordered pairs. (Contributed by Mario Carneiro, 3-May-2015.) |
Ref | Expression |
---|---|
opelvvg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ (V × V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3461 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | elex 3461 | . 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V) | |
3 | opelxpi 5668 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 ∈ (V × V)) | |
4 | 1, 2, 3 | syl2an 596 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ (V × V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 Vcvv 3443 〈cop 4590 × cxp 5629 |
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 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-opab 5166 df-xp 5637 |
This theorem is referenced by: relsnopg 5757 isof1oopb 7266 opvtxfv 27800 opiedgfv 27803 gonafv 33772 sat1el2xp 33801 opelvvdif 36651 brxrn 36768 |
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