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Theorem elxp8 36786
Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp7 8022. (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 8019 . . 3 (𝐴 ∈ (𝐵 × 𝐶) → (1st𝐴) ∈ 𝐵)
2 ssv 4002 . . . . 5 𝐵 ⊆ V
3 ssid 4000 . . . . 5 𝐶𝐶
4 xpss12 5687 . . . . 5 ((𝐵 ⊆ V ∧ 𝐶𝐶) → (𝐵 × 𝐶) ⊆ (V × 𝐶))
52, 3, 4mp2an 691 . . . 4 (𝐵 × 𝐶) ⊆ (V × 𝐶)
65sseli 3974 . . 3 (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ (V × 𝐶))
71, 6jca 511 . 2 (𝐴 ∈ (𝐵 × 𝐶) → ((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)))
8 xpss 5688 . . . . 5 (V × 𝐶) ⊆ (V × V)
98sseli 3974 . . . 4 (𝐴 ∈ (V × 𝐶) → 𝐴 ∈ (V × V))
109adantl 481 . . 3 (((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (V × V))
11 xp2nd 8020 . . . 4 (𝐴 ∈ (V × 𝐶) → (2nd𝐴) ∈ 𝐶)
1211anim2i 616 . . 3 (((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)) → ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))
13 elxp7 8022 . . 3 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
1410, 12, 13sylanbrc 582 . 2 (((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (𝐵 × 𝐶))
157, 14impbii 208 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wcel 2099  Vcvv 3469  wss 3944   × cxp 5670  cfv 6542  1st c1st 7985  2nd c2nd 7986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fv 6550  df-1st 7987  df-2nd 7988
This theorem is referenced by:  finxpsuclem  36812
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