Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ordge1n0im | GIF version |
Description: An ordinal greater than or equal to 1 is nonzero. (Contributed by Jim Kingdon, 26-Jun-2019.) |
Ref | Expression |
---|---|
ordge1n0im | ⊢ (Ord 𝐴 → (1o ⊆ 𝐴 → 𝐴 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordgt0ge1 6397 | . 2 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴)) | |
2 | ne0i 3413 | . 2 ⊢ (∅ ∈ 𝐴 → 𝐴 ≠ ∅) | |
3 | 1, 2 | syl6bir 163 | 1 ⊢ (Ord 𝐴 → (1o ⊆ 𝐴 → 𝐴 ≠ ∅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2135 ≠ wne 2334 ⊆ wss 3114 ∅c0 3407 Ord word 4337 1oc1o 6371 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-nul 4105 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-v 2726 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-nul 3408 df-pw 3558 df-sn 3579 df-uni 3787 df-tr 4078 df-iord 4341 df-on 4343 df-suc 4346 df-1o 6378 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |