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Mirrors > Home > ILE Home > Th. List > ordgt0ge1 | GIF version |
Description: Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.) |
Ref | Expression |
---|---|
ordgt0ge1 | ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1𝑜 ⊆ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elon 4147 | . . 3 ⊢ ∅ ∈ On | |
2 | ordelsuc 4249 | . . 3 ⊢ ((∅ ∈ On ∧ Ord 𝐴) → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴)) | |
3 | 1, 2 | mpan 414 | . 2 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴)) |
4 | df-1o 6024 | . . 3 ⊢ 1𝑜 = suc ∅ | |
5 | 4 | sseq1i 3023 | . 2 ⊢ (1𝑜 ⊆ 𝐴 ↔ suc ∅ ⊆ 𝐴) |
6 | 3, 5 | syl6bbr 196 | 1 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1𝑜 ⊆ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∈ wcel 1433 ⊆ wss 2973 ∅c0 3251 Ord word 4117 Oncon0 4118 suc csuc 4120 1𝑜c1o 6017 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-nul 3904 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-uni 3602 df-tr 3876 df-iord 4121 df-on 4123 df-suc 4126 df-1o 6024 |
This theorem is referenced by: ordge1n0im 6042 archnqq 6607 |
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