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Mirrors > Home > NFE 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 2862 | . 2 ⊢ x ∈ V | |
2 | ne0i 3556 | . 2 ⊢ (x ∈ V → V ≠ ∅) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ V ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1710 ≠ wne 2516 Vcvv 2859 ∅c0 3550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-nul 3551 |
This theorem is referenced by: uniintsn 3963 enpw 6087 2p1e3c 6156 ce0addcnnul 6179 ce2 6192 ncvsq 6256 |
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