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Theorem 2nd1st 8008
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 7998 . . . . 5 (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
21sneqd 4635 . . . 4 (𝐴 ∈ (𝐵 × 𝐶) → {𝐴} = {⟨(1st𝐴), (2nd𝐴)⟩})
32cnveqd 5868 . . 3 (𝐴 ∈ (𝐵 × 𝐶) → {𝐴} = {⟨(1st𝐴), (2nd𝐴)⟩})
43unieqd 4916 . 2 (𝐴 ∈ (𝐵 × 𝐶) → {𝐴} = {⟨(1st𝐴), (2nd𝐴)⟩})
5 opswap 6218 . 2 {⟨(1st𝐴), (2nd𝐴)⟩} = ⟨(2nd𝐴), (1st𝐴)⟩
64, 5eqtrdi 2788 1 (𝐴 ∈ (𝐵 × 𝐶) → {𝐴} = ⟨(2nd𝐴), (1st𝐴)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  {csn 4623  cop 4629   cuni 4902   × cxp 5668  ccnv 5669  cfv 6533  1st c1st 7957  2nd c2nd 7958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pr 5421  ax-un 7709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-iota 6485  df-fun 6535  df-fv 6541  df-1st 7959  df-2nd 7960
This theorem is referenced by:  fcnvgreu  31831  gsumhashmul  32143
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