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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 5484 | . 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 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 |
| 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 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 |
| This theorem is referenced by: eqvinop 5492 opelxp 5721 fsn 7155 opiota 8084 canthwe 10691 ltresr 11180 mat1dimelbas 22477 fmucndlem 24300 hgt750lemb 34671 diblsmopel 41173 cdlemn7 41205 dihordlem7 41216 xihopellsmN 41256 dihopellsm 41257 dihpN 41338 |
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