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Mirrors > Home > ILE Home > Th. List > vn0 | GIF version |
Description: The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.) |
Ref | Expression |
---|---|
vn0 | ⊢ V ≠ ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2759 | . 2 ⊢ 𝑥 ∈ V | |
2 | ne0i 3449 | . 2 ⊢ (𝑥 ∈ V → V ≠ ∅) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ V ≠ ∅ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 ≠ wne 2360 Vcvv 2756 ∅c0 3442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-v 2758 df-dif 3151 df-nul 3443 |
This theorem is referenced by: (None) |
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