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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 6131 | . 2 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1𝑜 ⊆ 𝐴)) | |
2 | ne0i 3275 | . 2 ⊢ (∅ ∈ 𝐴 → 𝐴 ≠ ∅) | |
3 | 1, 2 | syl6bir 162 | 1 ⊢ (Ord 𝐴 → (1𝑜 ⊆ 𝐴 → 𝐴 ≠ ∅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1434 ≠ wne 2249 ⊆ wss 2984 ∅c0 3269 Ord word 4153 1𝑜c1o 6106 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-nul 3930 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-v 2614 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-uni 3628 df-tr 3902 df-iord 4157 df-on 4159 df-suc 4162 df-1o 6113 |
This theorem is referenced by: (None) |
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