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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 7726. (Contributed by ML, 19-Oct-2020.) |
Ref | Expression |
---|---|
elxp8 | ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xp1st 7723 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1st ‘𝐴) ∈ 𝐵) | |
2 | ssv 3993 | . . . . 5 ⊢ 𝐵 ⊆ V | |
3 | ssid 3991 | . . . . 5 ⊢ 𝐶 ⊆ 𝐶 | |
4 | xpss12 5572 | . . . . 5 ⊢ ((𝐵 ⊆ V ∧ 𝐶 ⊆ 𝐶) → (𝐵 × 𝐶) ⊆ (V × 𝐶)) | |
5 | 2, 3, 4 | mp2an 690 | . . . 4 ⊢ (𝐵 × 𝐶) ⊆ (V × 𝐶) |
6 | 5 | sseli 3965 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ (V × 𝐶)) |
7 | 1, 6 | jca 514 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶))) |
8 | xpss 5573 | . . . . 5 ⊢ (V × 𝐶) ⊆ (V × V) | |
9 | 8 | sseli 3965 | . . . 4 ⊢ (𝐴 ∈ (V × 𝐶) → 𝐴 ∈ (V × V)) |
10 | 9 | adantl 484 | . . 3 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (V × V)) |
11 | xp2nd 7724 | . . . 4 ⊢ (𝐴 ∈ (V × 𝐶) → (2nd ‘𝐴) ∈ 𝐶) | |
12 | 11 | anim2i 618 | . . 3 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) |
13 | elxp7 7726 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) | |
14 | 10, 12, 13 | sylanbrc 585 | . 2 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (𝐵 × 𝐶)) |
15 | 7, 14 | impbii 211 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 × cxp 5555 ‘cfv 6357 1st c1st 7689 2nd c2nd 7690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fv 6365 df-1st 7691 df-2nd 7692 |
This theorem is referenced by: finxpsuclem 34680 |
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