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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 3570 | . . 3 ⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵}) | |
2 | 1 | neeq1d 2303 | . 2 ⊢ (𝑥 = 𝐴 → ({𝑥, 𝐵} ≠ ∅ ↔ {𝐴, 𝐵} ≠ ∅)) |
3 | vex 2663 | . . 3 ⊢ 𝑥 ∈ V | |
4 | 3 | prnz 3615 | . 2 ⊢ {𝑥, 𝐵} ≠ ∅ |
5 | 2, 4 | vtoclg 2720 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐵} ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∈ wcel 1465 ≠ wne 2285 ∅c0 3333 {cpr 3498 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-v 2662 df-dif 3043 df-un 3045 df-nul 3334 df-sn 3503 df-pr 3504 |
This theorem is referenced by: 0nelop 4140 |
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