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Mirrors > Home > ILE Home > Th. List > prnzg | GIF version |
Description: A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.) |
Ref | Expression |
---|---|
prnzg | ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐵} ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq1 3600 | . . 3 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵}) | |
2 | 1 | neeq1d 2326 | . 2 ⊢ (𝑥 = 𝐴 → ({𝑥, 𝐵} ≠ ∅ ↔ {𝐴, 𝐵} ≠ ∅)) |
3 | vex 2689 | . . 3 ⊢ 𝑥 ∈ V | |
4 | 3 | prnz 3645 | . 2 ⊢ {𝑥, 𝐵} ≠ ∅ |
5 | 2, 4 | vtoclg 2746 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐵} ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ≠ wne 2308 ∅c0 3363 {cpr 3528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-v 2688 df-dif 3073 df-un 3075 df-nul 3364 df-sn 3533 df-pr 3534 |
This theorem is referenced by: 0nelop 4170 |
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