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Theorem 2nd1st 5748
Description: Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)
Assertion
Ref Expression
2nd1st (A (B × 𝐶) → {A} = ⟨(2ndA), (1stA)⟩)

Proof of Theorem 2nd1st
StepHypRef Expression
1 1st2nd2 5743 . . . . 5 (A (B × 𝐶) → A = ⟨(1stA), (2ndA)⟩)
21sneqd 3380 . . . 4 (A (B × 𝐶) → {A} = {⟨(1stA), (2ndA)⟩})
32cnveqd 4454 . . 3 (A (B × 𝐶) → {A} = {⟨(1stA), (2ndA)⟩})
43unieqd 3582 . 2 (A (B × 𝐶) → {A} = {⟨(1stA), (2ndA)⟩})
5 1stexg 5736 . . 3 (A (B × 𝐶) → (1stA) V)
6 2ndexg 5737 . . 3 (A (B × 𝐶) → (2ndA) V)
7 opswapg 4750 . . 3 (((1stA) V (2ndA) V) → {⟨(1stA), (2ndA)⟩} = ⟨(2ndA), (1stA)⟩)
85, 6, 7syl2anc 391 . 2 (A (B × 𝐶) → {⟨(1stA), (2ndA)⟩} = ⟨(2ndA), (1stA)⟩)
94, 8eqtrd 2069 1 (A (B × 𝐶) → {A} = ⟨(2ndA), (1stA)⟩)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  Vcvv 2551  {csn 3367  cop 3370   cuni 3571   × cxp 4286  ccnv 4287  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: (None)
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