elxp8 - Mathbox for ML

	Mathbox for ML		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > elxp8		Structured version Visualization version GIF version

Theorem elxp8 38437

Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp7 8046. (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 8043	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1^st ‘𝐴) ∈ 𝐵)
2		ssv 3937	. . . . 5 ⊢ 𝐵 ⊆ V
3		ssid 3935	. . . . 5 ⊢ 𝐶 ⊆ 𝐶
4		xpss12 5670	. . . . 5 ⊢ ((𝐵 ⊆ V ∧ 𝐶 ⊆ 𝐶) → (𝐵 × 𝐶) ⊆ (V × 𝐶))
5	2, 3, 4	mp2an 705	. . . 4 ⊢ (𝐵 × 𝐶) ⊆ (V × 𝐶)
6	5	sseli 3909	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ (V × 𝐶))
7	1, 6	jca 521	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ((1^st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)))
8		xpss 5671	. . . . 5 ⊢ (V × 𝐶) ⊆ (V × V)
9	8	sseli 3909	. . . 4 ⊢ (𝐴 ∈ (V × 𝐶) → 𝐴 ∈ (V × V))
10	9	adantl 487	. . 3 ⊢ (((1^st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (V × V))
11		xp2nd 8044	. . . 4 ⊢ (𝐴 ∈ (V × 𝐶) → (2^nd ‘𝐴) ∈ 𝐶)
12	11	anim2i 629	. . 3 ⊢ (((1^st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)) → ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶))
13		elxp7 8046	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶)))
14	10, 12, 13	sylanbrc 595	. 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

This proof depends on syntax axioms: ↔ wb 209 ∧ wa 401 ∈ wcel 2146 Vcvv 3454 ⊆ wss 3900 × cxp 5653 ‘cfv 6547 1^st c1st 8009 2^nd c2nd 8010

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pr 5395 ax-un 7761

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-iota 6501 df-fun 6549 df-fv 6555 df-1st 8011 df-2nd 8012

This theorem is used by: finxpsuclem 38463

W3C validator