| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > onn0 | Structured version Visualization version GIF version | ||
| Description: The class of all ordinal numbers is not empty. (Contributed by NM, 17-Sep-1995.) |
| Ref | Expression |
|---|---|
| onn0 | ⊢ On ≠ ∅ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0elon 6389 | . 2 ⊢ ∅ ∈ On | |
| 2 | 1 | ne0ii 4309 | 1 ⊢ On ≠ ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: ≠ wne 2926 ∅c0 4298 Oncon0 6334 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-ext 2702 ax-nul 5263 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2066 df-clab 2709 df-cleq 2722 df-clel 2804 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-dif 3919 df-un 3921 df-ss 3933 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-br 5110 df-tr 5217 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-ord 6337 df-on 6338 |
| This theorem is referenced by: limon 7813 |
| Copyright terms: Public domain | W3C validator |