![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
Mirrors > Home > ILE Home > Th. List > ordge1n0im | Structured version 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 5928 | . 2 ⊢ (Ord A → (∅ ∈ A ↔ 1𝑜 ⊆ A)) | |
2 | ne0i 3206 | . 2 ⊢ (∅ ∈ A → A ≠ ∅) | |
3 | 1, 2 | syl6bir 153 | 1 ⊢ (Ord A → (1𝑜 ⊆ A → A ≠ ∅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1375 ≠ wne 2187 ⊆ wss 2893 ∅c0 3200 Ord word 4046 1𝑜c1o 5904 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 532 ax-in2 533 ax-io 617 ax-5 1316 ax-7 1317 ax-gen 1318 ax-ie1 1364 ax-ie2 1365 ax-8 1377 ax-10 1378 ax-11 1379 ax-i12 1380 ax-bnd 1381 ax-4 1382 ax-17 1401 ax-i9 1405 ax-ial 1410 ax-i5r 1411 ax-ext 2005 ax-nul 3856 |
This theorem depends on definitions: df-bi 110 df-tru 1231 df-nf 1330 df-sb 1629 df-clab 2010 df-cleq 2016 df-clel 2019 df-nfc 2150 df-ne 2189 df-ral 2288 df-rex 2289 df-v 2536 df-dif 2896 df-un 2898 df-in 2900 df-ss 2907 df-nul 3201 df-pw 3335 df-sn 3355 df-uni 3554 df-tr 3828 df-iord 4050 df-on 4052 df-suc 4055 df-1o 5911 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |