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Theorem elxp8 33349
Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp7 7245. (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 7242 . . 3 (𝐴 ∈ (𝐵 × 𝐶) → (1st𝐴) ∈ 𝐵)
2 ssv 3658 . . . . 5 𝐵 ⊆ V
3 ssid 3657 . . . . 5 𝐶𝐶
4 xpss12 5158 . . . . 5 ((𝐵 ⊆ V ∧ 𝐶𝐶) → (𝐵 × 𝐶) ⊆ (V × 𝐶))
52, 3, 4mp2an 708 . . . 4 (𝐵 × 𝐶) ⊆ (V × 𝐶)
65sseli 3632 . . 3 (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ (V × 𝐶))
71, 6jca 553 . 2 (𝐴 ∈ (𝐵 × 𝐶) → ((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)))
8 xpss 5159 . . . . 5 (V × 𝐶) ⊆ (V × V)
98sseli 3632 . . . 4 (𝐴 ∈ (V × 𝐶) → 𝐴 ∈ (V × V))
109adantl 481 . . 3 (((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (V × V))
11 xp2nd 7243 . . . 4 (𝐴 ∈ (V × 𝐶) → (2nd𝐴) ∈ 𝐶)
1211anim2i 592 . . 3 (((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)) → ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))
13 elxp7 7245 . . 3 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
1410, 12, 13sylanbrc 699 . 2 (((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (𝐵 × 𝐶))
157, 14impbii 199 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wcel 2030  Vcvv 3231  wss 3607   × cxp 5141  cfv 5926  1st c1st 7208  2nd c2nd 7209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fv 5934  df-1st 7210  df-2nd 7211
This theorem is referenced by:  finxpsuclem  33364
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