![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 4129 | . . 3 ⊢ ∅ ∈ On | |
2 | ordelsuc 4231 | . . 3 ⊢ ((∅ ∈ On ∧ Ord 𝐴) → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴)) | |
3 | 1, 2 | mpan 400 | . 2 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴)) |
4 | df-1o 6001 | . . 3 ⊢ 1𝑜 = suc ∅ | |
5 | 4 | sseq1i 2969 | . 2 ⊢ (1𝑜 ⊆ 𝐴 ↔ suc ∅ ⊆ 𝐴) |
6 | 3, 5 | syl6bbr 187 | 1 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1𝑜 ⊆ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∈ wcel 1393 ⊆ wss 2917 ∅c0 3224 Ord word 4099 Oncon0 4100 suc csuc 4102 1𝑜c1o 5994 |
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 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-nul 3883 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-uni 3581 df-tr 3855 df-iord 4103 df-on 4105 df-suc 4108 df-1o 6001 |
This theorem is referenced by: ordge1n0im 6019 archnqq 6515 |
Copyright terms: Public domain | W3C validator |