MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opeqsn Structured version   Visualization version   GIF version

Theorem opeqsn 5107
Description: Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) (Proof shortened by AV, 16-Jun-2022.)
Hypotheses
Ref Expression
opeqsn.1 𝐴 ∈ V
opeqsn.2 𝐵 ∈ V
opeqsn.3 𝐶 ∈ V
Assertion
Ref Expression
opeqsn (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵𝐶 = {𝐴}))

Proof of Theorem opeqsn
StepHypRef Expression
1 opeqsn.1 . . . 4 𝐴 ∈ V
2 opeqsn.2 . . . 4 𝐵 ∈ V
31, 2dfop 4544 . . 3 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
43eqeq1i 2757 . 2 (⟨𝐴, 𝐵⟩ = {𝐶} ↔ {{𝐴}, {𝐴, 𝐵}} = {𝐶})
5 snex 5049 . . 3 {𝐴} ∈ V
6 prex 5050 . . 3 {𝐴, 𝐵} ∈ V
75, 6preqsn 4532 . 2 ({{𝐴}, {𝐴, 𝐵}} = {𝐶} ↔ ({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶))
8 eqcom 2759 . . . . 5 ({𝐴} = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = {𝐴})
91, 2preqsn 4532 . . . . 5 ({𝐴, 𝐵} = {𝐴} ↔ (𝐴 = 𝐵𝐵 = 𝐴))
10 eqcom 2759 . . . . . . 7 (𝐵 = 𝐴𝐴 = 𝐵)
1110anbi2i 732 . . . . . 6 ((𝐴 = 𝐵𝐵 = 𝐴) ↔ (𝐴 = 𝐵𝐴 = 𝐵))
12 anidm 679 . . . . . 6 ((𝐴 = 𝐵𝐴 = 𝐵) ↔ 𝐴 = 𝐵)
1311, 12bitri 264 . . . . 5 ((𝐴 = 𝐵𝐵 = 𝐴) ↔ 𝐴 = 𝐵)
148, 9, 133bitri 286 . . . 4 ({𝐴} = {𝐴, 𝐵} ↔ 𝐴 = 𝐵)
1514anbi1i 733 . . 3 (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶))
16 dfsn2 4326 . . . . . . 7 {𝐴} = {𝐴, 𝐴}
17 preq2 4405 . . . . . . 7 (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
1816, 17syl5req 2799 . . . . . 6 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
1918eqeq1d 2754 . . . . 5 (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶 ↔ {𝐴} = 𝐶))
20 eqcom 2759 . . . . 5 ({𝐴} = 𝐶𝐶 = {𝐴})
2119, 20syl6bb 276 . . . 4 (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶𝐶 = {𝐴}))
2221pm5.32i 672 . . 3 ((𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵𝐶 = {𝐴}))
2315, 22bitri 264 . 2 (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵𝐶 = {𝐴}))
244, 7, 233bitri 286 1 (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵𝐶 = {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1624  wcel 2131  Vcvv 3332  {csn 4313  {cpr 4315  cop 4319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320
This theorem is referenced by:  snopeqop  5109  propeqop  5110  relop  5420
  Copyright terms: Public domain W3C validator