Theorem preqsn 3777

Description: Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)

Hypotheses

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

Assertion

Ref	Expression
preqsn	⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶))

Proof of Theorem preqsn

Step	Hyp	Ref	Expression
1		dfsn2 3608	. . 3 ⊢ {𝐶} = {𝐶, 𝐶}
2	1	eqeq2i 2188	. 2 ⊢ ({𝐴, 𝐵} = {𝐶} ↔ {𝐴, 𝐵} = {𝐶, 𝐶})
3		preqsn.1	. . . 4 ⊢ 𝐴 ∈ V
4		preqsn.2	. . . 4 ⊢ 𝐵 ∈ V
5		preqsn.3	. . . 4 ⊢ 𝐶 ∈ V
6	3, 4, 5, 5	preq12b 3772	. . 3 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)))
7		oridm 757	. . . 4 ⊢ (((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))
8		eqtr3 2197	. . . . . 6 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵)
9		simpr 110	. . . . . 6 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶)
10	8, 9	jca 306	. . . . 5 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → (𝐴 = 𝐵 ∧ 𝐵 = 𝐶))
11		eqtr 2195	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐶)
12		simpr 110	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶)
13	11, 12	jca 306	. . . . 5 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))
14	10, 13	impbii 126	. . . 4 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶))
15	7, 14	bitri 184	. . 3 ⊢ (((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶))
16	6, 15	bitri 184	. 2 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶))
17	2, 16	bitri 184	1 ⊢ ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∨ wo 708   = wceq 1353   ∈ wcel 2148  Vcvv 2739  {csn 3594  {cpr 3595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601

This theorem is referenced by:  opeqsn  4254  relop  4779