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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 6460 | . 2 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴)) | |
2 | ne0i 3444 | . 2 ⊢ (∅ ∈ 𝐴 → 𝐴 ≠ ∅) | |
3 | 1, 2 | biimtrrdi 164 | 1 ⊢ (Ord 𝐴 → (1o ⊆ 𝐴 → 𝐴 ≠ ∅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2160 ≠ wne 2360 ⊆ wss 3144 ∅c0 3437 Ord word 4380 1oc1o 6434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-nul 4144 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-uni 3825 df-tr 4117 df-iord 4384 df-on 4386 df-suc 4389 df-1o 6441 |
This theorem is referenced by: (None) |
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