Theorem elpreqpr 4872

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 4655	. 2 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
2		elex 3481	. 2 ⊢ (𝐴 ∈ {𝐵, 𝐶} → 𝐴 ∈ V)
3		elpreqprlem 4871	. . . . 5 ⊢ (𝐵 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥})
4		eleq1 2813	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
5		preq1 4741	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → {𝐴, 𝑥} = {𝐵, 𝑥})
6	5	eqeq2d 2736	. . . . . . 7 ⊢ (𝐴 = 𝐵 → ({𝐵, 𝐶} = {𝐴, 𝑥} ↔ {𝐵, 𝐶} = {𝐵, 𝑥}))
7	6	exbidv 1916	. . . . . 6 ⊢ (𝐴 = 𝐵 → (∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥} ↔ ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥}))
8	4, 7	imbi12d 343	. . . . 5 ⊢ (𝐴 = 𝐵 → ((𝐴 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥}) ↔ (𝐵 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥})))
9	3, 8	mpbiri 257	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥}))
10	9	imp 405	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ∈ V) → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})
11		elpreqprlem 4871	. . . . . 6 ⊢ (𝐶 ∈ V → ∃𝑥{𝐶, 𝐵} = {𝐶, 𝑥})
12		prcom 4740	. . . . . . . 8 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶}
13	12	eqeq1i 2730	. . . . . . 7 ⊢ ({𝐶, 𝐵} = {𝐶, 𝑥} ↔ {𝐵, 𝐶} = {𝐶, 𝑥})
14	13	exbii 1842	. . . . . 6 ⊢ (∃𝑥{𝐶, 𝐵} = {𝐶, 𝑥} ↔ ∃𝑥{𝐵, 𝐶} = {𝐶, 𝑥})
15	11, 14	sylib 217	. . . . 5 ⊢ (𝐶 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐶, 𝑥})
16		eleq1 2813	. . . . . 6 ⊢ (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V))
17		preq1 4741	. . . . . . . 8 ⊢ (𝐴 = 𝐶 → {𝐴, 𝑥} = {𝐶, 𝑥})
18	17	eqeq2d 2736	. . . . . . 7 ⊢ (𝐴 = 𝐶 → ({𝐵, 𝐶} = {𝐴, 𝑥} ↔ {𝐵, 𝐶} = {𝐶, 𝑥}))
19	18	exbidv 1916	. . . . . 6 ⊢ (𝐴 = 𝐶 → (∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥} ↔ ∃𝑥{𝐵, 𝐶} = {𝐶, 𝑥}))
20	16, 19	imbi12d 343	. . . . 5 ⊢ (𝐴 = 𝐶 → ((𝐴 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥}) ↔ (𝐶 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐶, 𝑥})))
21	15, 20	mpbiri 257	. . . 4 ⊢ (𝐴 = 𝐶 → (𝐴 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥}))
22	21	imp 405	. . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐴 ∈ V) → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})
23	10, 22	jaoian 954	. 2 ⊢ (((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) ∧ 𝐴 ∈ V) → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})
24	1, 2, 23	syl2anc 582	1 ⊢ (𝐴 ∈ {𝐵, 𝐶} → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})

Syntax hints:   → wi 4   ∨ wo 845   = wceq 1533  ∃wex 1773   ∈ wcel 2098  Vcvv 3461  {cpr 4634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3463  df-dif 3949  df-un 3951  df-nul 4325  df-sn 4633  df-pr 4635

This theorem is referenced by:  elpreqprb  4873