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Mirrors > Home > MPE Home > Th. List > Mathboxes > elxp8 | Structured version Visualization version GIF version |
Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp7 7706. (Contributed by ML, 19-Oct-2020.) |
Ref | Expression |
---|---|
elxp8 | ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xp1st 7703 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1st ‘𝐴) ∈ 𝐵) | |
2 | ssv 3939 | . . . . 5 ⊢ 𝐵 ⊆ V | |
3 | ssid 3937 | . . . . 5 ⊢ 𝐶 ⊆ 𝐶 | |
4 | xpss12 5534 | . . . . 5 ⊢ ((𝐵 ⊆ V ∧ 𝐶 ⊆ 𝐶) → (𝐵 × 𝐶) ⊆ (V × 𝐶)) | |
5 | 2, 3, 4 | mp2an 691 | . . . 4 ⊢ (𝐵 × 𝐶) ⊆ (V × 𝐶) |
6 | 5 | sseli 3911 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ (V × 𝐶)) |
7 | 1, 6 | jca 515 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶))) |
8 | xpss 5535 | . . . . 5 ⊢ (V × 𝐶) ⊆ (V × V) | |
9 | 8 | sseli 3911 | . . . 4 ⊢ (𝐴 ∈ (V × 𝐶) → 𝐴 ∈ (V × V)) |
10 | 9 | adantl 485 | . . 3 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (V × V)) |
11 | xp2nd 7704 | . . . 4 ⊢ (𝐴 ∈ (V × 𝐶) → (2nd ‘𝐴) ∈ 𝐶) | |
12 | 11 | anim2i 619 | . . 3 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) |
13 | elxp7 7706 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) | |
14 | 10, 12, 13 | sylanbrc 586 | . 2 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (𝐵 × 𝐶)) |
15 | 7, 14 | impbii 212 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 × cxp 5517 ‘cfv 6324 1st c1st 7669 2nd c2nd 7670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fv 6332 df-1st 7671 df-2nd 7672 |
This theorem is referenced by: finxpsuclem 34814 |
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