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Theorem elxp8 37337

Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp7 8065. (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 8062	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1^st ‘𝐴) ∈ 𝐵)
2		ssv 4033	. . . . 5 ⊢ 𝐵 ⊆ V
3		ssid 4031	. . . . 5 ⊢ 𝐶 ⊆ 𝐶
4		xpss12 5715	. . . . 5 ⊢ ((𝐵 ⊆ V ∧ 𝐶 ⊆ 𝐶) → (𝐵 × 𝐶) ⊆ (V × 𝐶))
5	2, 3, 4	mp2an 691	. . . 4 ⊢ (𝐵 × 𝐶) ⊆ (V × 𝐶)
6	5	sseli 4004	. . 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 5716	. . . . 5 ⊢ (V × 𝐶) ⊆ (V × V)
9	8	sseli 4004	. . . 4 ⊢ (𝐴 ∈ (V × 𝐶) → 𝐴 ∈ (V × V))
10	9	adantl 481	. . 3 ⊢ (((1^st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (V × V))
11		xp2nd 8063	. . . 4 ⊢ (𝐴 ∈ (V × 𝐶) → (2^nd ‘𝐴) ∈ 𝐶)
12	11	anim2i 616	. . 3 ⊢ (((1^st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶))
13		elxp7 8065	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶)))
14	10, 12, 13	sylanbrc 582	. 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 2108 Vcvv 3488 ⊆ wss 3976 × cxp 5698 ‘cfv 6573 1^st c1st 8028 2^nd c2nd 8029

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-iota 6525 df-fun 6575 df-fv 6581 df-1st 8030 df-2nd 8031

This theorem is referenced by: finxpsuclem 37363

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