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Theorem elxp8 35185

Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp7 7749. (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 7746	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1^st ‘𝐴) ∈ 𝐵)
2		ssv 3901	. . . . 5 ⊢ 𝐵 ⊆ V
3		ssid 3899	. . . . 5 ⊢ 𝐶 ⊆ 𝐶
4		xpss12 5540	. . . . 5 ⊢ ((𝐵 ⊆ V ∧ 𝐶 ⊆ 𝐶) → (𝐵 × 𝐶) ⊆ (V × 𝐶))
5	2, 3, 4	mp2an 692	. . . 4 ⊢ (𝐵 × 𝐶) ⊆ (V × 𝐶)
6	5	sseli 3873	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ (V × 𝐶))
7	1, 6	jca 515	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ((1^st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)))
8		xpss 5541	. . . . 5 ⊢ (V × 𝐶) ⊆ (V × V)
9	8	sseli 3873	. . . 4 ⊢ (𝐴 ∈ (V × 𝐶) → 𝐴 ∈ (V × V))
10	9	adantl 485	. . 3 ⊢ (((1^st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (V × V))
11		xp2nd 7747	. . . 4 ⊢ (𝐴 ∈ (V × 𝐶) → (2^nd ‘𝐴) ∈ 𝐶)
12	11	anim2i 620	. . 3 ⊢ (((1^st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶))
13		elxp7 7749	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶)))
14	10, 12, 13	sylanbrc 586	. 2 ⊢ (((1^st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (𝐵 × 𝐶))
15	7, 14	impbii 212	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1^st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 209 ∧ wa 399 ∈ wcel 2114 Vcvv 3398 ⊆ wss 3843 × cxp 5523 ‘cfv 6339 1^st c1st 7712 2^nd c2nd 7713

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7479

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-iota 6297 df-fun 6341 df-fv 6347 df-1st 7714 df-2nd 7715

This theorem is referenced by: finxpsuclem 35211

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