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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 5224 | . 2 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | |
4 | 1, 2, 3 | mp2an 680 | 1 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 Vcvv 3408 〈cop 4441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pr 5182 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-rab 3090 df-v 3410 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 |
This theorem is referenced by: eqvinop 5231 opelxp 5439 fsn 6718 opiota 7563 canthwe 9869 ltresr 10358 mat1dimelbas 20799 fmucndlem 22618 hgt750lemb 31607 diblsmopel 37789 cdlemn7 37821 dihordlem7 37832 xihopellsmN 37872 dihopellsm 37873 dihpN 37954 |
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