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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 4407 | . . 3 ⊢ ∅ ∈ On | |
2 | ordelsuc 4519 | . . 3 ⊢ ((∅ ∈ On ∧ Ord 𝐴) → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴)) | |
3 | 1, 2 | mpan 424 | . 2 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ suc ∅ ⊆ 𝐴)) |
4 | df-1o 6436 | . . 3 ⊢ 1o = suc ∅ | |
5 | 4 | sseq1i 3196 | . 2 ⊢ (1o ⊆ 𝐴 ↔ suc ∅ ⊆ 𝐴) |
6 | 3, 5 | bitr4di 198 | 1 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∈ wcel 2160 ⊆ wss 3144 ∅c0 3437 Ord word 4377 Oncon0 4378 suc csuc 4380 1oc1o 6429 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-nul 4144 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-uni 3825 df-tr 4117 df-iord 4381 df-on 4383 df-suc 4386 df-1o 6436 |
This theorem is referenced by: ordge1n0im 6456 archnqq 7441 |
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