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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 4874 | . 2 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | |
4 | 1, 2, 3 | mp2an 704 | 1 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 |
This theorem is referenced by: eqvinop 4881 opelxp 5070 fsn 6308 opiota 7118 canthwe 9352 ltresr 9840 mat1dimelbas 20096 fmucndlem 21905 diblsmopel 35478 cdlemn7 35510 dihordlem7 35521 xihopellsmN 35561 dihopellsm 35562 dihpN 35643 |
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