Theorem opswap 6230

Description: Swap the members of an ordered pair. (Contributed by NM, 14-Dec-2008.) (Revised by Mario Carneiro, 30-Aug-2015.)

Assertion

Ref	Expression
opswap	⊢ ∪ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩

Proof of Theorem opswap

Step	Hyp	Ref	Expression
1		cnvsng 6224	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
2	1	unieqd 4885	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{⟨𝐴, 𝐵⟩} = ∪ {⟨𝐵, 𝐴⟩})
3		opex 5445	. . . 4 ⊢ ⟨𝐵, 𝐴⟩ ∈ V
4	3	unisn 4891	. . 3 ⊢ ∪ {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴⟩
5	2, 4	eqtrdi 2814	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)
6		uni0 4901	. . 3 ⊢ ∪ ∅ = ∅
7		opprc 4861	. . . . . . 7 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
8	7	sneqd 4601	. . . . . 6 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {∅})
9	8	cnveqd 5861	. . . . 5 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{⟨𝐴, 𝐵⟩} = ◡{∅})
10		cnvsn0 6211	. . . . 5 ⊢ ◡{∅} = ∅
11	9, 10	eqtrdi 2814	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{⟨𝐴, 𝐵⟩} = ∅)
12	11	unieqd 4885	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{⟨𝐴, 𝐵⟩} = ∪ ∅)
13		ancom 465	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V))
14		opprc 4861	. . . 4 ⊢ (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
15	13, 14	sylnbi 333	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
16	6, 12, 15	3eqtr4a 2824	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)
17	5, 16	pm2.61i 184	1 ⊢ ∪ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩

Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ∅c0 4286  {csn 4589  ⟨cop 4595  ∪ cuni 4872  ^◡ccnv 5660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672

This theorem is referenced by:  2nd1st  8031  cnvf1olem  8101  brtpos  8227  dftpos4  8237  tpostpos  8238  xpcomco  9051  fsumcnv  15820  fprodcnv  16033  gsumcom2  20040  txswaphmeolem  23961  swapf1  50070  swapf2  50072