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Theorem elxp8 37687

Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp7 7977. (Contributed by ML, 19-Oct-2020.)

**Assertion**
Ref	Expression
elxp8	⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1^st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)))

**Proof of Theorem elxp8**
Step	Hyp	Ref	Expression
1		xp1st 7974	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1^st ‘𝐴) ∈ 𝐵)
2		ssv 3946	. . . . 5 ⊢ 𝐵 ⊆ V
3		ssid 3944	. . . . 5 ⊢ 𝐶 ⊆ 𝐶
4		xpss12 5646	. . . . 5 ⊢ ((𝐵 ⊆ V ∧ 𝐶 ⊆ 𝐶) → (𝐵 × 𝐶) ⊆ (V × 𝐶))
5	2, 3, 4	mp2an 693	. . . 4 ⊢ (𝐵 × 𝐶) ⊆ (V × 𝐶)
6	5	sseli 3917	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ (V × 𝐶))
7	1, 6	jca 511	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ((1^st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)))
8		xpss 5647	. . . . 5 ⊢ (V × 𝐶) ⊆ (V × V)
9	8	sseli 3917	. . . 4 ⊢ (𝐴 ∈ (V × 𝐶) → 𝐴 ∈ (V × V))
10	9	adantl 481	. . 3 ⊢ (((1^st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (V × V))
11		xp2nd 7975	. . . 4 ⊢ (𝐴 ∈ (V × 𝐶) → (2^nd ‘𝐴) ∈ 𝐶)
12	11	anim2i 618	. . 3 ⊢ (((1^st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶))
13		elxp7 7977	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶)))
14	10, 12, 13	sylanbrc 584	. 2 ⊢ (((1^st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (𝐵 × 𝐶))
15	7, 14	impbii 209	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1^st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2114 Vcvv 3429 ⊆ wss 3889 × cxp 5629 ‘cfv 6498 1^st c1st 7940 2^nd c2nd 7941

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pr 5375 ax-un 7689

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-iota 6454 df-fun 6500 df-fv 6506 df-1st 7942 df-2nd 7943

This theorem is referenced by: finxpsuclem 37713

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