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Mirrors > Home > MPE Home > Th. List > elpreqprlem | Structured version Visualization version GIF version |
Description: Lemma for elpreqpr 4867. (Contributed by Scott Fenton, 7-Dec-2020.) (Revised by AV, 9-Dec-2020.) |
Ref | Expression |
---|---|
elpreqprlem | ⊢ (𝐵 ∈ 𝑉 → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2732 | . . . 4 ⊢ {𝐵, 𝐶} = {𝐵, 𝐶} | |
2 | preq2 4738 | . . . . . 6 ⊢ (𝑥 = 𝐶 → {𝐵, 𝑥} = {𝐵, 𝐶}) | |
3 | 2 | eqeq2d 2743 | . . . . 5 ⊢ (𝑥 = 𝐶 → ({𝐵, 𝐶} = {𝐵, 𝑥} ↔ {𝐵, 𝐶} = {𝐵, 𝐶})) |
4 | 3 | spcegv 3587 | . . . 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 4641 | . . . 4 ⊢ {𝐵} = {𝐵, 𝐵} | |
8 | preq2 4738 | . . . . . 6 ⊢ (𝑥 = 𝐵 → {𝐵, 𝑥} = {𝐵, 𝐵}) | |
9 | 8 | eqeq2d 2743 | . . . . 5 ⊢ (𝑥 = 𝐵 → ({𝐵} = {𝐵, 𝑥} ↔ {𝐵} = {𝐵, 𝐵})) |
10 | 9 | spcegv 3587 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → ({𝐵} = {𝐵, 𝐵} → ∃𝑥{𝐵} = {𝐵, 𝑥})) |
11 | 7, 10 | mpi 20 | . . 3 ⊢ (𝐵 ∈ 𝑉 → ∃𝑥{𝐵} = {𝐵, 𝑥}) |
12 | prprc2 4770 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → {𝐵, 𝐶} = {𝐵}) | |
13 | 12 | eqeq1d 2734 | . . . 4 ⊢ (¬ 𝐶 ∈ V → ({𝐵, 𝐶} = {𝐵, 𝑥} ↔ {𝐵} = {𝐵, 𝑥})) |
14 | 13 | exbidv 1924 | . . 3 ⊢ (¬ 𝐶 ∈ V → (∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥} ↔ ∃𝑥{𝐵} = {𝐵, 𝑥})) |
15 | 11, 14 | imbitrrid 245 | . 2 ⊢ (¬ 𝐶 ∈ V → (𝐵 ∈ 𝑉 → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥})) |
16 | 6, 15 | pm2.61i 182 | 1 ⊢ (𝐵 ∈ 𝑉 → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1541 ∃wex 1781 ∈ wcel 2106 Vcvv 3474 {csn 4628 {cpr 4630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-dif 3951 df-un 3953 df-nul 4323 df-sn 4629 df-pr 4631 |
This theorem is referenced by: elpreqpr 4867 |
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