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Theorem elxp7 5739
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4751. (Contributed by NM, 19-Aug-2006.)
Assertion
Ref Expression
elxp7 (A (B × 𝐶) ↔ (A (V × V) ((1stA) B (2ndA) 𝐶)))

Proof of Theorem elxp7
StepHypRef Expression
1 elex 2560 . 2 (A (B × 𝐶) → A V)
2 elex 2560 . . 3 (A (V × V) → A V)
32adantr 261 . 2 ((A (V × V) ((1stA) B (2ndA) 𝐶)) → A V)
4 1stexg 5736 . . . . . . 7 (A V → (1stA) V)
5 2ndexg 5737 . . . . . . 7 (A V → (2ndA) V)
64, 5jca 290 . . . . . 6 (A V → ((1stA) V (2ndA) V))
76biantrud 288 . . . . 5 (A V → (A = ⟨(1stA), (2ndA)⟩ ↔ (A = ⟨(1stA), (2ndA)⟩ ((1stA) V (2ndA) V))))
8 elxp6 5738 . . . . 5 (A (V × V) ↔ (A = ⟨(1stA), (2ndA)⟩ ((1stA) V (2ndA) V)))
97, 8syl6rbbr 188 . . . 4 (A V → (A (V × V) ↔ A = ⟨(1stA), (2ndA)⟩))
109anbi1d 438 . . 3 (A V → ((A (V × V) ((1stA) B (2ndA) 𝐶)) ↔ (A = ⟨(1stA), (2ndA)⟩ ((1stA) B (2ndA) 𝐶))))
11 elxp6 5738 . . 3 (A (B × 𝐶) ↔ (A = ⟨(1stA), (2ndA)⟩ ((1stA) B (2ndA) 𝐶)))
1210, 11syl6rbbr 188 . 2 (A V → (A (B × 𝐶) ↔ (A (V × V) ((1stA) B (2ndA) 𝐶))))
131, 3, 12pm5.21nii 619 1 (A (B × 𝐶) ↔ (A (V × V) ((1stA) B (2ndA) 𝐶)))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242   wcel 1390  Vcvv 2551  cop 3370   × cxp 4286  cfv 4845  1st c1st 5707  2nd c2nd 5708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-1st 5709  df-2nd 5710
This theorem is referenced by:  xp2  5741  unielxp  5742  1stconst  5784  2ndconst  5785
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