Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4354 | . 2 ⊢ Tr On | |
2 | df-on 4340 | . . . . 5 ⊢ On = {𝑥 ∣ Ord 𝑥} | |
3 | 2 | abeq2i 2275 | . . . 4 ⊢ (𝑥 ∈ On ↔ Ord 𝑥) |
4 | ordtr 4350 | . . . 4 ⊢ (Ord 𝑥 → Tr 𝑥) | |
5 | 3, 4 | sylbi 120 | . . 3 ⊢ (𝑥 ∈ On → Tr 𝑥) |
6 | 5 | rgen 2517 | . 2 ⊢ ∀𝑥 ∈ On Tr 𝑥 |
7 | dford3 4339 | . 2 ⊢ (Ord On ↔ (Tr On ∧ ∀𝑥 ∈ On Tr 𝑥)) | |
8 | 1, 6, 7 | mpbir2an 931 | 1 ⊢ Ord On |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2135 ∀wral 2442 Tr wtr 4074 Ord word 4334 Oncon0 4335 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-in 3117 df-ss 3124 df-uni 3784 df-tr 4075 df-iord 4338 df-on 4340 |
This theorem is referenced by: ssorduni 4458 limon 4484 onprc 4523 tfri1dALT 6310 rdgon 6345 |
Copyright terms: Public domain | W3C validator |