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Theorem elxp8 37349

Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp7 7959. (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 7956	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1^st ‘𝐴) ∈ 𝐵)
2		ssv 3960	. . . . 5 ⊢ 𝐵 ⊆ V
3		ssid 3958	. . . . 5 ⊢ 𝐶 ⊆ 𝐶
4		xpss12 5634	. . . . 5 ⊢ ((𝐵 ⊆ V ∧ 𝐶 ⊆ 𝐶) → (𝐵 × 𝐶) ⊆ (V × 𝐶))
5	2, 3, 4	mp2an 692	. . . 4 ⊢ (𝐵 × 𝐶) ⊆ (V × 𝐶)
6	5	sseli 3931	. . 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 5635	. . . . 5 ⊢ (V × 𝐶) ⊆ (V × V)
9	8	sseli 3931	. . . 4 ⊢ (𝐴 ∈ (V × 𝐶) → 𝐴 ∈ (V × V))
10	9	adantl 481	. . 3 ⊢ (((1^st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (V × V))
11		xp2nd 7957	. . . 4 ⊢ (𝐴 ∈ (V × 𝐶) → (2^nd ‘𝐴) ∈ 𝐶)
12	11	anim2i 617	. . 3 ⊢ (((1^st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶))
13		elxp7 7959	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶)))
14	10, 12, 13	sylanbrc 583	. 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 2109 Vcvv 3436 ⊆ wss 3903 × cxp 5617 ‘cfv 6482 1^st c1st 7922 2^nd c2nd 7923

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pr 5371 ax-un 7671

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3395 df-v 3438 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-iota 6438 df-fun 6484 df-fv 6490 df-1st 7924 df-2nd 7925

This theorem is referenced by: finxpsuclem 37375

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