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Theorem elxp8 34646
Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp7 7718. (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 7715 . . 3 (𝐴 ∈ (𝐵 × 𝐶) → (1st𝐴) ∈ 𝐵)
2 ssv 3990 . . . . 5 𝐵 ⊆ V
3 ssid 3988 . . . . 5 𝐶𝐶
4 xpss12 5564 . . . . 5 ((𝐵 ⊆ V ∧ 𝐶𝐶) → (𝐵 × 𝐶) ⊆ (V × 𝐶))
52, 3, 4mp2an 690 . . . 4 (𝐵 × 𝐶) ⊆ (V × 𝐶)
65sseli 3962 . . 3 (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ (V × 𝐶))
71, 6jca 514 . 2 (𝐴 ∈ (𝐵 × 𝐶) → ((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)))
8 xpss 5565 . . . . 5 (V × 𝐶) ⊆ (V × V)
98sseli 3962 . . . 4 (𝐴 ∈ (V × 𝐶) → 𝐴 ∈ (V × V))
109adantl 484 . . 3 (((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (V × V))
11 xp2nd 7716 . . . 4 (𝐴 ∈ (V × 𝐶) → (2nd𝐴) ∈ 𝐶)
1211anim2i 618 . . 3 (((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)) → ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))
13 elxp7 7718 . . 3 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
1410, 12, 13sylanbrc 585 . 2 (((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (𝐵 × 𝐶))
157, 14impbii 211 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398  wcel 2110  Vcvv 3494  wss 3935   × cxp 5547  cfv 6349  1st c1st 7681  2nd c2nd 7682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-iota 6308  df-fun 6351  df-fv 6357  df-1st 7683  df-2nd 7684
This theorem is referenced by:  finxpsuclem  34672
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