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Mirrors > Home > MPE Home > Th. List > opth2 | Structured version Visualization version GIF version |
Description: Ordered pair theorem. (Contributed by NM, 21-Sep-2014.) |
Ref | Expression |
---|---|
opth2.1 | ⊢ 𝐶 ∈ V |
opth2.2 | ⊢ 𝐷 ∈ V |
Ref | Expression |
---|---|
opth2 | ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opth2.1 | . 2 ⊢ 𝐶 ∈ V | |
2 | opth2.2 | . 2 ⊢ 𝐷 ∈ V | |
3 | opthg2 5489 | . 2 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | |
4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 〈cop 4636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 |
This theorem is referenced by: eqvinop 5497 opelxp 5724 fsn 7154 opiota 8082 canthwe 10688 ltresr 11177 mat1dimelbas 22492 fmucndlem 24315 hgt750lemb 34649 diblsmopel 41153 cdlemn7 41185 dihordlem7 41196 xihopellsmN 41236 dihopellsm 41237 dihpN 41318 |
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