Theorem elpreqprlem 4618

Description: Lemma for elpreqpr 4619. (Contributed by Scott Fenton, 7-Dec-2020.) (Revised by AV, 9-Dec-2020.)

Assertion

Ref	Expression
elpreqprlem	⊢ (𝐵 ∈ 𝑉 → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥})

Distinct variable groups:   𝑥,𝐵   𝑥,𝐶

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem elpreqprlem

Step	Hyp	Ref	Expression
1		eqid 2825	. . . 4 ⊢ {𝐵, 𝐶} = {𝐵, 𝐶}
2		preq2 4489	. . . . . 6 ⊢ (𝑥 = 𝐶 → {𝐵, 𝑥} = {𝐵, 𝐶})
3	2	eqeq2d 2835	. . . . 5 ⊢ (𝑥 = 𝐶 → ({𝐵, 𝐶} = {𝐵, 𝑥} ↔ {𝐵, 𝐶} = {𝐵, 𝐶}))
4	3	spcegv 3511	. . . 4 ⊢ (𝐶 ∈ V → ({𝐵, 𝐶} = {𝐵, 𝐶} → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥}))
5	1, 4	mpi 20	. . 3 ⊢ (𝐶 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥})
6	5	a1d 25	. 2 ⊢ (𝐶 ∈ V → (𝐵 ∈ 𝑉 → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥}))
7		dfsn2 4412	. . . 4 ⊢ {𝐵} = {𝐵, 𝐵}
8		preq2 4489	. . . . . 6 ⊢ (𝑥 = 𝐵 → {𝐵, 𝑥} = {𝐵, 𝐵})
9	8	eqeq2d 2835	. . . . 5 ⊢ (𝑥 = 𝐵 → ({𝐵} = {𝐵, 𝑥} ↔ {𝐵} = {𝐵, 𝐵}))
10	9	spcegv 3511	. . . 4 ⊢ (𝐵 ∈ 𝑉 → ({𝐵} = {𝐵, 𝐵} → ∃𝑥{𝐵} = {𝐵, 𝑥}))
11	7, 10	mpi 20	. . 3 ⊢ (𝐵 ∈ 𝑉 → ∃𝑥{𝐵} = {𝐵, 𝑥})
12		prprc2 4521	. . . . 5 ⊢ (¬ 𝐶 ∈ V → {𝐵, 𝐶} = {𝐵})
13	12	eqeq1d 2827	. . . 4 ⊢ (¬ 𝐶 ∈ V → ({𝐵, 𝐶} = {𝐵, 𝑥} ↔ {𝐵} = {𝐵, 𝑥}))
14	13	exbidv 2020	. . 3 ⊢ (¬ 𝐶 ∈ V → (∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥} ↔ ∃𝑥{𝐵} = {𝐵, 𝑥}))
15	11, 14	syl5ibr 238	. 2 ⊢ (¬ 𝐶 ∈ V → (𝐵 ∈ 𝑉 → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥}))
16	6, 15	pm2.61i 177	1 ⊢ (𝐵 ∈ 𝑉 → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥})

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1656  ∃wex 1878   ∈ wcel 2164  Vcvv 3414  {csn 4399  {cpr 4401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-v 3416  df-dif 3801  df-un 3803  df-nul 4147  df-sn 4400  df-pr 4402

This theorem is referenced by:  elpreqpr  4619