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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 5075 | . 2 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | |
4 | 1, 2, 3 | mp2an 672 | 1 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 Vcvv 3351 〈cop 4322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 |
This theorem is referenced by: eqvinop 5082 opelxp 5286 fsn 6545 opiota 7378 canthwe 9675 ltresr 10163 mat1dimelbas 20495 fmucndlem 22315 hgt750lemb 31074 diblsmopel 36981 cdlemn7 37013 dihordlem7 37024 xihopellsmN 37064 dihopellsm 37065 dihpN 37146 |
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