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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 4228 | . . 3 ⊢ ∅ ∈ On | |
2 | ordelsuc 4335 | . . 3 ⊢ ((∅ ∈ On ∧ Ord 𝐴) → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴)) | |
3 | 1, 2 | mpan 416 | . 2 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴)) |
4 | df-1o 6195 | . . 3 ⊢ 1o = suc ∅ | |
5 | 4 | sseq1i 3051 | . 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 1439 ⊆ wss 3000 ∅c0 3287 Ord word 4198 Oncon0 4199 suc csuc 4201 1oc1o 6188 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-nul 3971 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-uni 3660 df-tr 3943 df-iord 4202 df-on 4204 df-suc 4207 df-1o 6195 |
This theorem is referenced by: ordge1n0im 6214 archnqq 7037 |
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