Theorem preqr1g 3558

Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 3560. (Contributed by Jim Kingdon, 21-Sep-2018.)

Assertion

Ref	Expression
preqr1g	⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵))

Proof of Theorem preqr1g

Step	Hyp	Ref	Expression
1		prid1g 3496	. . . . . . 7 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴, 𝐶})
2		eleq2 2142	. . . . . . 7 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 ∈ {𝐴, 𝐶} ↔ 𝐴 ∈ {𝐵, 𝐶}))
3	1, 2	syl5ibcom 153	. . . . . 6 ⊢ (𝐴 ∈ V → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 ∈ {𝐵, 𝐶}))
4		elprg 3418	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
5	3, 4	sylibd 147	. . . . 5 ⊢ (𝐴 ∈ V → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
6	5	adantr 270	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
7	6	imp 122	. . 3 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
8		prid1g 3496	. . . . . . 7 ⊢ (𝐵 ∈ V → 𝐵 ∈ {𝐵, 𝐶})
9		eleq2 2142	. . . . . . 7 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 ∈ {𝐴, 𝐶} ↔ 𝐵 ∈ {𝐵, 𝐶}))
10	8, 9	syl5ibrcom 155	. . . . . 6 ⊢ (𝐵 ∈ V → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐵 ∈ {𝐴, 𝐶}))
11		elprg 3418	. . . . . 6 ⊢ (𝐵 ∈ V → (𝐵 ∈ {𝐴, 𝐶} ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)))
12	10, 11	sylibd 147	. . . . 5 ⊢ (𝐵 ∈ V → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)))
13	12	adantl 271	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)))
14	13	imp 122	. . 3 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → (𝐵 = 𝐴 ∨ 𝐵 = 𝐶))
15		eqcom 2083	. . 3 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
16		eqeq2 2090	. . 3 ⊢ (𝐴 = 𝐶 → (𝐵 = 𝐴 ↔ 𝐵 = 𝐶))
17	7, 14, 15, 16	oplem1 916	. 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → 𝐴 = 𝐵)
18	17	ex 113	1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵))

Syntax hints:   → wi 4   ∧ wa 102   ∨ wo 661   = wceq 1284   ∈ wcel 1433  Vcvv 2601  {cpr 3399

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405

This theorem is referenced by:  preqr2g  3559