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Theorem 2nd1st 7173
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 7165 . . . . 5 (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
21sneqd 4167 . . . 4 (𝐴 ∈ (𝐵 × 𝐶) → {𝐴} = {⟨(1st𝐴), (2nd𝐴)⟩})
32cnveqd 5268 . . 3 (𝐴 ∈ (𝐵 × 𝐶) → {𝐴} = {⟨(1st𝐴), (2nd𝐴)⟩})
43unieqd 4419 . 2 (𝐴 ∈ (𝐵 × 𝐶) → {𝐴} = {⟨(1st𝐴), (2nd𝐴)⟩})
5 opswap 5591 . 2 {⟨(1st𝐴), (2nd𝐴)⟩} = ⟨(2nd𝐴), (1st𝐴)⟩
64, 5syl6eq 2671 1 (𝐴 ∈ (𝐵 × 𝐶) → {𝐴} = ⟨(2nd𝐴), (1st𝐴)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  {csn 4155  cop 4161   cuni 4409   × cxp 5082  ccnv 5083  cfv 5857  1st c1st 7126  2nd c2nd 7127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-iota 5820  df-fun 5859  df-fv 5865  df-1st 7128  df-2nd 7129
This theorem is referenced by:  fcnvgreu  29356
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