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Mirrors > Home > MPE Home > Th. List > prnzgOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of prnzg 4254 as of 23-Jul-2021. (Contributed by FL, 19-Sep-2011.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
prnzgOLD | ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐵} ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq1 4212 | . . 3 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵}) | |
2 | 1 | neeq1d 2841 | . 2 ⊢ (𝑥 = 𝐴 → ({𝑥, 𝐵} ≠ ∅ ↔ {𝐴, 𝐵} ≠ ∅)) |
3 | vex 3176 | . . 3 ⊢ 𝑥 ∈ V | |
4 | 3 | prnz 4253 | . 2 ⊢ {𝑥, 𝐵} ≠ ∅ |
5 | 2, 4 | vtoclg 3239 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐵} ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 {cpr 4127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-dif 3543 df-un 3545 df-nul 3875 df-sn 4126 df-pr 4128 |
This theorem is referenced by: (None) |
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