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Theorem elxp8 33759
Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp7 7468. (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 7465 . . 3 (𝐴 ∈ (𝐵 × 𝐶) → (1st𝐴) ∈ 𝐵)
2 ssv 3850 . . . . 5 𝐵 ⊆ V
3 ssid 3848 . . . . 5 𝐶𝐶
4 xpss12 5361 . . . . 5 ((𝐵 ⊆ V ∧ 𝐶𝐶) → (𝐵 × 𝐶) ⊆ (V × 𝐶))
52, 3, 4mp2an 683 . . . 4 (𝐵 × 𝐶) ⊆ (V × 𝐶)
65sseli 3823 . . 3 (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ (V × 𝐶))
71, 6jca 507 . 2 (𝐴 ∈ (𝐵 × 𝐶) → ((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)))
8 xpss 5362 . . . . 5 (V × 𝐶) ⊆ (V × V)
98sseli 3823 . . . 4 (𝐴 ∈ (V × 𝐶) → 𝐴 ∈ (V × V))
109adantl 475 . . 3 (((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (V × V))
11 xp2nd 7466 . . . 4 (𝐴 ∈ (V × 𝐶) → (2nd𝐴) ∈ 𝐶)
1211anim2i 610 . . 3 (((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)) → ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶))
13 elxp7 7468 . . 3 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
1410, 12, 13sylanbrc 578 . 2 (((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)) → 𝐴 ∈ (𝐵 × 𝐶))
157, 14impbii 201 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1st𝐴) ∈ 𝐵𝐴 ∈ (V × 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386  wcel 2164  Vcvv 3414  wss 3798   × cxp 5344  cfv 6127  1st c1st 7431  2nd c2nd 7432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-iota 6090  df-fun 6129  df-fv 6135  df-1st 7433  df-2nd 7434
This theorem is referenced by:  finxpsuclem  33774
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