Theorem preqr1g 3746

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 3748. (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 3680	. . . . . . 7 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴, 𝐶})
2		eleq2 2230	. . . . . . 7 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 ∈ {𝐴, 𝐶} ↔ 𝐴 ∈ {𝐵, 𝐶}))
3	1, 2	syl5ibcom 154	. . . . . 6 ⊢ (𝐴 ∈ V → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 ∈ {𝐵, 𝐶}))
4		elprg 3596	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
5	3, 4	sylibd 148	. . . . 5 ⊢ (𝐴 ∈ V → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
6	5	adantr 274	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
7	6	imp 123	. . 3 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
8		prid1g 3680	. . . . . . 7 ⊢ (𝐵 ∈ V → 𝐵 ∈ {𝐵, 𝐶})
9		eleq2 2230	. . . . . . 7 ⊢ ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 ∈ {𝐴, 𝐶} ↔ 𝐵 ∈ {𝐵, 𝐶}))
10	8, 9	syl5ibrcom 156	. . . . . 6 ⊢ (𝐵 ∈ V → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐵 ∈ {𝐴, 𝐶}))
11		elprg 3596	. . . . . 6 ⊢ (𝐵 ∈ V → (𝐵 ∈ {𝐴, 𝐶} ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)))
12	10, 11	sylibd 148	. . . . 5 ⊢ (𝐵 ∈ V → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)))
13	12	adantl 275	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐶} = {𝐵, 𝐶} → (𝐵 = 𝐴 ∨ 𝐵 = 𝐶)))
14	13	imp 123	. . 3 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → (𝐵 = 𝐴 ∨ 𝐵 = 𝐶))
15		eqcom 2167	. . 3 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
16		eqeq2 2175	. . 3 ⊢ (𝐴 = 𝐶 → (𝐵 = 𝐴 ↔ 𝐵 = 𝐶))
17	7, 14, 15, 16	oplem1 965	. 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ {𝐴, 𝐶} = {𝐵, 𝐶}) → 𝐴 = 𝐵)
18	17	ex 114	1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 698   = wceq 1343   ∈ wcel 2136  Vcvv 2726  {cpr 3577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583

This theorem is referenced by:  preqr2g  3747