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Mirrors > Home > ILE Home > Th. List > ordon | GIF version |
Description: The class of all ordinal numbers is ordinal. Proposition 7.12 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. (Contributed by NM, 17-May-1994.) |
Ref | Expression |
---|---|
ordon | ⊢ Ord On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tron 4274 | . 2 ⊢ Tr On | |
2 | df-on 4260 | . . . . 5 ⊢ On = {𝑥 ∣ Ord 𝑥} | |
3 | 2 | abeq2i 2228 | . . . 4 ⊢ (𝑥 ∈ On ↔ Ord 𝑥) |
4 | ordtr 4270 | . . . 4 ⊢ (Ord 𝑥 → Tr 𝑥) | |
5 | 3, 4 | sylbi 120 | . . 3 ⊢ (𝑥 ∈ On → Tr 𝑥) |
6 | 5 | rgen 2462 | . 2 ⊢ ∀𝑥 ∈ On Tr 𝑥 |
7 | dford3 4259 | . 2 ⊢ (Ord On ↔ (Tr On ∧ ∀𝑥 ∈ On Tr 𝑥)) | |
8 | 1, 6, 7 | mpbir2an 911 | 1 ⊢ Ord On |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1465 ∀wral 2393 Tr wtr 3996 Ord word 4254 Oncon0 4255 |
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-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 |
This theorem depends on definitions: df-bi 116 df-3an 949 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-in 3047 df-ss 3054 df-uni 3707 df-tr 3997 df-iord 4258 df-on 4260 |
This theorem is referenced by: ssorduni 4373 limon 4399 onprc 4437 tfri1dALT 6216 rdgon 6251 |
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