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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) ↔ ((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)))

Proof of Theorem elxp8
StepHypRef Expression
1 xp1st 7746 . . 3 (𝐴 ∈ (𝐵 × 𝐶) → (1st𝐴) ∈ 𝐵)
2 ssv 3901 . . . . 5 𝐵 ⊆ V
3 ssid 3899 . . . . 5 𝐶𝐶
4 xpss12 5540 . . . . 5 ((𝐵 ⊆ V ∧ 𝐶𝐶) → (𝐵 × 𝐶) ⊆ (V × 𝐶))
52, 3, 4mp2an 692 . . . 4 (𝐵 × 𝐶) ⊆ (V × 𝐶)
65sseli 3873 . . 3 (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ (V × 𝐶))
71, 6jca 515 . 2 (𝐴 ∈ (𝐵 × 𝐶) → ((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)))
8 xpss 5541 . . . . 5 (V × 𝐶) ⊆ (V × V)
98sseli 3873 . . . 4 (𝐴 ∈ (V × 𝐶) → 𝐴 ∈ (V × V))
109adantl 485 . . 3 (((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (V × V))
11 xp2nd 7747 . . . 4 (𝐴 ∈ (V × 𝐶) → (2nd𝐴) ∈ 𝐶)
1211anim2i 620 . . 3 (((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)) → ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))
13 elxp7 7749 . . 3 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
1410, 12, 13sylanbrc 586 . 2 (((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (𝐵 × 𝐶))
157, 14impbii 212 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wcel 2114  Vcvv 3398  wss 3843   × cxp 5523  cfv 6339  1st c1st 7712  2nd 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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