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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 4378 | . 2 ⊢ Tr On | |
2 | df-on 4364 | . . . . 5 ⊢ On = {𝑥 ∣ Ord 𝑥} | |
3 | 2 | abeq2i 2288 | . . . 4 ⊢ (𝑥 ∈ On ↔ Ord 𝑥) |
4 | ordtr 4374 | . . . 4 ⊢ (Ord 𝑥 → Tr 𝑥) | |
5 | 3, 4 | sylbi 121 | . . 3 ⊢ (𝑥 ∈ On → Tr 𝑥) |
6 | 5 | rgen 2530 | . 2 ⊢ ∀𝑥 ∈ On Tr 𝑥 |
7 | dford3 4363 | . 2 ⊢ (Ord On ↔ (Tr On ∧ ∀𝑥 ∈ On Tr 𝑥)) | |
8 | 1, 6, 7 | mpbir2an 942 | 1 ⊢ Ord On |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2148 ∀wral 2455 Tr wtr 4098 Ord word 4358 Oncon0 4359 |
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-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-in 3135 df-ss 3142 df-uni 3808 df-tr 4099 df-iord 4362 df-on 4364 |
This theorem is referenced by: ssorduni 4482 limon 4508 onprc 4547 tfri1dALT 6345 rdgon 6380 |
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