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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 4095 | . . 3 ⊢ ∅ ∈ On | |
2 | ordelsuc 4197 | . . 3 ⊢ ((∅ ∈ On ∧ Ord A) → (∅ ∈ A ↔ suc ∅ ⊆ A)) | |
3 | 1, 2 | mpan 400 | . 2 ⊢ (Ord A → (∅ ∈ A ↔ suc ∅ ⊆ A)) |
4 | df-1o 5940 | . . 3 ⊢ 1𝑜 = suc ∅ | |
5 | 4 | sseq1i 2963 | . 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 1390 ⊆ wss 2911 ∅c0 3218 Ord word 4065 Oncon0 4066 suc csuc 4068 1𝑜c1o 5933 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-nul 3874 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-uni 3572 df-tr 3846 df-iord 4069 df-on 4071 df-suc 4074 df-1o 5940 |
This theorem is referenced by: ordge1n0im 5958 archnqq 6400 |
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