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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 4284 | . . 3 ⊢ ∅ ∈ On | |
2 | ordelsuc 4391 | . . 3 ⊢ ((∅ ∈ On ∧ Ord 𝐴) → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴)) | |
3 | 1, 2 | mpan 420 | . 2 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴)) |
4 | df-1o 6281 | . . 3 ⊢ 1o = suc ∅ | |
5 | 4 | sseq1i 3093 | . 2 ⊢ (1o ⊆ 𝐴 ↔ suc ∅ ⊆ 𝐴) |
6 | 3, 5 | syl6bbr 197 | 1 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∈ wcel 1465 ⊆ wss 3041 ∅c0 3333 Ord word 4254 Oncon0 4255 suc csuc 4257 1oc1o 6274 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-nul 4024 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-uni 3707 df-tr 3997 df-iord 4258 df-on 4260 df-suc 4263 df-1o 6281 |
This theorem is referenced by: ordge1n0im 6301 archnqq 7193 |
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