Theorem elpreqpr 4797

Description: Equality and membership rule for pairs. (Contributed by Scott Fenton, 7-Dec-2020.)

Assertion

Ref	Expression
elpreqpr	⊢ (𝐴 ∈ {𝐵, 𝐶} → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem elpreqpr

Step	Hyp	Ref	Expression
1		elpri 4589	. 2 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
2		elex 3512	. 2 ⊢ (𝐴 ∈ {𝐵, 𝐶} → 𝐴 ∈ V)
3		elpreqprlem 4796	. . . . 5 ⊢ (𝐵 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥})
4		eleq1 2900	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
5		preq1 4669	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → {𝐴, 𝑥} = {𝐵, 𝑥})
6	5	eqeq2d 2832	. . . . . . 7 ⊢ (𝐴 = 𝐵 → ({𝐵, 𝐶} = {𝐴, 𝑥} ↔ {𝐵, 𝐶} = {𝐵, 𝑥}))
7	6	exbidv 1922	. . . . . 6 ⊢ (𝐴 = 𝐵 → (∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥} ↔ ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥}))
8	4, 7	imbi12d 347	. . . . 5 ⊢ (𝐴 = 𝐵 → ((𝐴 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥}) ↔ (𝐵 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥})))
9	3, 8	mpbiri 260	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥}))
10	9	imp 409	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ∈ V) → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})
11		elpreqprlem 4796	. . . . . 6 ⊢ (𝐶 ∈ V → ∃𝑥{𝐶, 𝐵} = {𝐶, 𝑥})
12		prcom 4668	. . . . . . . 8 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶}
13	12	eqeq1i 2826	. . . . . . 7 ⊢ ({𝐶, 𝐵} = {𝐶, 𝑥} ↔ {𝐵, 𝐶} = {𝐶, 𝑥})
14	13	exbii 1848	. . . . . 6 ⊢ (∃𝑥{𝐶, 𝐵} = {𝐶, 𝑥} ↔ ∃𝑥{𝐵, 𝐶} = {𝐶, 𝑥})
15	11, 14	sylib 220	. . . . 5 ⊢ (𝐶 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐶, 𝑥})
16		eleq1 2900	. . . . . 6 ⊢ (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V))
17		preq1 4669	. . . . . . . 8 ⊢ (𝐴 = 𝐶 → {𝐴, 𝑥} = {𝐶, 𝑥})
18	17	eqeq2d 2832	. . . . . . 7 ⊢ (𝐴 = 𝐶 → ({𝐵, 𝐶} = {𝐴, 𝑥} ↔ {𝐵, 𝐶} = {𝐶, 𝑥}))
19	18	exbidv 1922	. . . . . 6 ⊢ (𝐴 = 𝐶 → (∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥} ↔ ∃𝑥{𝐵, 𝐶} = {𝐶, 𝑥}))
20	16, 19	imbi12d 347	. . . . 5 ⊢ (𝐴 = 𝐶 → ((𝐴 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥}) ↔ (𝐶 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐶, 𝑥})))
21	15, 20	mpbiri 260	. . . 4 ⊢ (𝐴 = 𝐶 → (𝐴 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥}))
22	21	imp 409	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐴 ∈ V) → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})
23	10, 22	jaoian 953	. 2 ⊢ (((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) ∧ 𝐴 ∈ V) → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})
24	1, 2, 23	syl2anc 586	1 ⊢ (𝐴 ∈ {𝐵, 𝐶} → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})

Syntax hints:   → wi 4   ∨ wo 843   = wceq 1537  ∃wex 1780   ∈ wcel 2114  Vcvv 3494  {cpr 4569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-dif 3939  df-un 3941  df-nul 4292  df-sn 4568  df-pr 4570

This theorem is referenced by:  elpreqprb  4798