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Theorem opeqsn 4016
Description: Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)
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 3575 . . 3 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
43eqeq1i 2063 . 2 (⟨𝐴, 𝐵⟩ = {𝐶} ↔ {{𝐴}, {𝐴, 𝐵}} = {𝐶})
5 snexgOLD 3962 . . . 4 (𝐴 ∈ V → {𝐴} ∈ V)
61, 5ax-mp 7 . . 3 {𝐴} ∈ V
7 prexgOLD 3973 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
81, 2, 7mp2an 410 . . 3 {𝐴, 𝐵} ∈ V
9 opeqsn.3 . . 3 𝐶 ∈ V
106, 8, 9preqsn 3573 . 2 ({{𝐴}, {𝐴, 𝐵}} = {𝐶} ↔ ({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶))
11 eqcom 2058 . . . . 5 ({𝐴} = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = {𝐴})
121, 2, 1preqsn 3573 . . . . 5 ({𝐴, 𝐵} = {𝐴} ↔ (𝐴 = 𝐵𝐵 = 𝐴))
13 eqcom 2058 . . . . . . 7 (𝐵 = 𝐴𝐴 = 𝐵)
1413anbi2i 438 . . . . . 6 ((𝐴 = 𝐵𝐵 = 𝐴) ↔ (𝐴 = 𝐵𝐴 = 𝐵))
15 anidm 382 . . . . . 6 ((𝐴 = 𝐵𝐴 = 𝐵) ↔ 𝐴 = 𝐵)
1614, 15bitri 177 . . . . 5 ((𝐴 = 𝐵𝐵 = 𝐴) ↔ 𝐴 = 𝐵)
1711, 12, 163bitri 199 . . . 4 ({𝐴} = {𝐴, 𝐵} ↔ 𝐴 = 𝐵)
1817anbi1i 439 . . 3 (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶))
19 dfsn2 3416 . . . . . . 7 {𝐴} = {𝐴, 𝐴}
20 preq2 3475 . . . . . . 7 (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
2119, 20syl5req 2101 . . . . . 6 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
2221eqeq1d 2064 . . . . 5 (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶 ↔ {𝐴} = 𝐶))
23 eqcom 2058 . . . . 5 ({𝐴} = 𝐶𝐶 = {𝐴})
2422, 23syl6bb 189 . . . 4 (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶𝐶 = {𝐴}))
2524pm5.32i 435 . . 3 ((𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵𝐶 = {𝐴}))
2618, 25bitri 177 . 2 (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵𝐶 = {𝐴}))
274, 10, 263bitri 199 1 (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵𝐶 = {𝐴}))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102   = wceq 1259  wcel 1409  Vcvv 2574  {csn 3402  {cpr 3403  cop 3405
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411
This theorem is referenced by:  relop  4513
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