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Mirrors > Home > ILE Home > Th. List > eqop2 | GIF version |
Description: Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.) |
Ref | Expression |
---|---|
eqop2.1 | ⊢ 𝐵 ∈ V |
eqop2.2 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
eqop2 | ⊢ (𝐴 = 〈𝐵, 𝐶〉 ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqop2.1 | . . . 4 ⊢ 𝐵 ∈ V | |
2 | eqop2.2 | . . . 4 ⊢ 𝐶 ∈ V | |
3 | 1, 2 | opelvv 4676 | . . 3 ⊢ 〈𝐵, 𝐶〉 ∈ (V × V) |
4 | eleq1 2240 | . . 3 ⊢ (𝐴 = 〈𝐵, 𝐶〉 → (𝐴 ∈ (V × V) ↔ 〈𝐵, 𝐶〉 ∈ (V × V))) | |
5 | 3, 4 | mpbiri 168 | . 2 ⊢ (𝐴 = 〈𝐵, 𝐶〉 → 𝐴 ∈ (V × V)) |
6 | eqop 6177 | . 2 ⊢ (𝐴 ∈ (V × V) → (𝐴 = 〈𝐵, 𝐶〉 ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) | |
7 | 5, 6 | biadan2 456 | 1 ⊢ (𝐴 = 〈𝐵, 𝐶〉 ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 Vcvv 2737 〈cop 3595 × cxp 4624 ‘cfv 5216 1st c1st 6138 2nd c2nd 6139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fo 5222 df-fv 5224 df-1st 6140 df-2nd 6141 |
This theorem is referenced by: (None) |
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