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Mirrors > Home > ILE Home > Th. List > elxp7 | GIF version |
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5021. (Contributed by NM, 19-Aug-2006.) |
Ref | Expression |
---|---|
elxp7 | ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2692 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ V) | |
2 | elex 2692 | . . 3 ⊢ (𝐴 ∈ (V × V) → 𝐴 ∈ V) | |
3 | 2 | adantr 274 | . 2 ⊢ ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → 𝐴 ∈ V) |
4 | 1stexg 6058 | . . . . . . 7 ⊢ (𝐴 ∈ V → (1st ‘𝐴) ∈ V) | |
5 | 2ndexg 6059 | . . . . . . 7 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) ∈ V) | |
6 | 4, 5 | jca 304 | . . . . . 6 ⊢ (𝐴 ∈ V → ((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V)) |
7 | 6 | biantrud 302 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ↔ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V)))) |
8 | elxp6 6060 | . . . . 5 ⊢ (𝐴 ∈ (V × V) ↔ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V))) | |
9 | 7, 8 | syl6rbbr 198 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ (V × V) ↔ 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉)) |
10 | 9 | anbi1d 460 | . . 3 ⊢ (𝐴 ∈ V → ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) ↔ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))) |
11 | elxp6 6060 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) | |
12 | 10, 11 | syl6rbbr 198 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))) |
13 | 1, 3, 12 | pm5.21nii 693 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 Vcvv 2681 〈cop 3525 × cxp 4532 ‘cfv 5118 1st c1st 6029 2nd c2nd 6030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fo 5124 df-fv 5126 df-1st 6031 df-2nd 6032 |
This theorem is referenced by: xp2 6064 unielxp 6065 1stconst 6111 2ndconst 6112 f1od2 6125 |
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