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Mirrors > Home > MPE Home > Th. List > elpreqprlem | Structured version Visualization version GIF version |
Description: Lemma for elpreqpr 4791. (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 2821 | . . . 4 ⊢ {𝐵, 𝐶} = {𝐵, 𝐶} | |
2 | preq2 4664 | . . . . . 6 ⊢ (𝑥 = 𝐶 → {𝐵, 𝑥} = {𝐵, 𝐶}) | |
3 | 2 | eqeq2d 2832 | . . . . 5 ⊢ (𝑥 = 𝐶 → ({𝐵, 𝐶} = {𝐵, 𝑥} ↔ {𝐵, 𝐶} = {𝐵, 𝐶})) |
4 | 3 | spcegv 3597 | . . . 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 4574 | . . . 4 ⊢ {𝐵} = {𝐵, 𝐵} | |
8 | preq2 4664 | . . . . . 6 ⊢ (𝑥 = 𝐵 → {𝐵, 𝑥} = {𝐵, 𝐵}) | |
9 | 8 | eqeq2d 2832 | . . . . 5 ⊢ (𝑥 = 𝐵 → ({𝐵} = {𝐵, 𝑥} ↔ {𝐵} = {𝐵, 𝐵})) |
10 | 9 | spcegv 3597 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → ({𝐵} = {𝐵, 𝐵} → ∃𝑥{𝐵} = {𝐵, 𝑥})) |
11 | 7, 10 | mpi 20 | . . 3 ⊢ (𝐵 ∈ 𝑉 → ∃𝑥{𝐵} = {𝐵, 𝑥}) |
12 | prprc2 4696 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → {𝐵, 𝐶} = {𝐵}) | |
13 | 12 | eqeq1d 2823 | . . . 4 ⊢ (¬ 𝐶 ∈ V → ({𝐵, 𝐶} = {𝐵, 𝑥} ↔ {𝐵} = {𝐵, 𝑥})) |
14 | 13 | exbidv 1918 | . . 3 ⊢ (¬ 𝐶 ∈ V → (∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥} ↔ ∃𝑥{𝐵} = {𝐵, 𝑥})) |
15 | 11, 14 | syl5ibr 248 | . 2 ⊢ (¬ 𝐶 ∈ V → (𝐵 ∈ 𝑉 → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥})) |
16 | 6, 15 | pm2.61i 184 | 1 ⊢ (𝐵 ∈ 𝑉 → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∃wex 1776 ∈ wcel 2110 Vcvv 3495 {csn 4561 {cpr 4563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3497 df-dif 3939 df-un 3941 df-nul 4292 df-sn 4562 df-pr 4564 |
This theorem is referenced by: elpreqpr 4791 |
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