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Mirrors > Home > ILE Home > Th. List > ordon | Unicode 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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tron 4344 | . 2 | |
2 | df-on 4330 | . . . . 5 | |
3 | 2 | abeq2i 2268 | . . . 4 |
4 | ordtr 4340 | . . . 4 | |
5 | 3, 4 | sylbi 120 | . . 3 |
6 | 5 | rgen 2510 | . 2 |
7 | dford3 4329 | . 2 | |
8 | 1, 6, 7 | mpbir2an 927 | 1 |
Colors of variables: wff set class |
Syntax hints: wcel 2128 wral 2435 wtr 4064 word 4324 con0 4325 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-in 3108 df-ss 3115 df-uni 3775 df-tr 4065 df-iord 4328 df-on 4330 |
This theorem is referenced by: ssorduni 4448 limon 4474 onprc 4513 tfri1dALT 6300 rdgon 6335 |
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