Theorem opeqsn 2808

Description: Equivalence for an ordered pair equal to a singleton.

Hypotheses

Ref	Expression
opeqsn.1	⊢ A ∈ V
opeqsn.2	⊢ B ∈ V
opeqsn.3	⊢ C ∈ V

Assertion

Ref	Expression
opeqsn	⊢ (⟨A, B⟩ = {C} ↔ (A = B ⋀ C = {A}))

Proof of Theorem opeqsn

Step	Hyp	Ref	Expression
1		df-op 2420	. . 3 ⊢ ⟨A, B⟩ = {{A}, {A, B}}
2	1	eqeq1i 1485	. 2 ⊢ (⟨A, B⟩ = {C} ↔ {{A}, {A, B}} = {C})
3		snex 2756	. . 3 ⊢ {A} ∈ V
4		prex 2787	. . 3 ⊢ {A, B} ∈ V
5		opeqsn.3	. . 3 ⊢ C ∈ V
6	3, 4, 5	preqsn 2490	. 2 ⊢ ({{A}, {A, B}} = {C} ↔ ({A} = {A, B} ⋀ {A, B} = C))
7		eqcom 1480	. . . . 5 ⊢ ({A} = {A, B} ↔ {A, B} = {A})
8		opeqsn.1	. . . . . 6 ⊢ A ∈ V
9		opeqsn.2	. . . . . 6 ⊢ B ∈ V
10	8, 9, 8	preqsn 2490	. . . . 5 ⊢ ({A, B} = {A} ↔ (A = B ⋀ B = A))
11		eqcom 1480	. . . . . . 7 ⊢ (B = A ↔ A = B)
12	11	anbi2i 482	. . . . . 6 ⊢ ((A = B ⋀ B = A) ↔ (A = B ⋀ A = B))
13		anidm 434	. . . . . 6 ⊢ ((A = B ⋀ A = B) ↔ A = B)
14	12, 13	bitr 173	. . . . 5 ⊢ ((A = B ⋀ B = A) ↔ A = B)
15	7, 10, 14	3bitr 177	. . . 4 ⊢ ({A} = {A, B} ↔ A = B)
16	15	anbi1i 483	. . 3 ⊢ (({A} = {A, B} ⋀ {A, B} = C) ↔ (A = B ⋀ {A, B} = C))
17		preq2 2453	. . . . . . 7 ⊢ (A = B → {A, A} = {A, B})
18		dfsn2 2424	. . . . . . 7 ⊢ {A} = {A, A}
19	17, 18	syl5req 1523	. . . . . 6 ⊢ (A = B → {A, B} = {A})
20	19	eqeq1d 1486	. . . . 5 ⊢ (A = B → ({A, B} = C ↔ {A} = C))
21		eqcom 1480	. . . . 5 ⊢ ({A} = C ↔ C = {A})
22	20, 21	syl6bb 538	. . . 4 ⊢ (A = B → ({A, B} = C ↔ C = {A}))
23	22	pm5.32i 647	. . 3 ⊢ ((A = B ⋀ {A, B} = C) ↔ (A = B ⋀ C = {A}))
24	16, 23	bitr 173	. 2 ⊢ (({A} = {A, B} ⋀ {A, B} = C) ↔ (A = B ⋀ C = {A}))
25	2, 6, 24	3bitr 177	1 ⊢ (⟨A, B⟩ = {C} ↔ (A = B ⋀ C = {A}))

Syntax hints:   ↔ wb 146   ⋀ wa 223   = wceq 958   ∈ wcel 960  Vcvv 1814  {csn 2413  {cpr 2414  ⟨cop 2415

This theorem is referenced by:  relop 3281

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420

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