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Mirrors > Home > ILE Home > Th. List > limon | GIF version |
Description: The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.) |
Ref | Expression |
---|---|
limon | ⊢ Lim On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordon 4340 | . 2 ⊢ Ord On | |
2 | 0elon 4252 | . 2 ⊢ ∅ ∈ On | |
3 | unon 4365 | . . 3 ⊢ ∪ On = On | |
4 | 3 | eqcomi 2104 | . 2 ⊢ On = ∪ On |
5 | dflim2 4230 | . 2 ⊢ (Lim On ↔ (Ord On ∧ ∅ ∈ On ∧ On = ∪ On)) | |
6 | 1, 2, 4, 5 | mpbir3an 1131 | 1 ⊢ Lim On |
Colors of variables: wff set class |
Syntax hints: = wceq 1299 ∈ wcel 1448 ∅c0 3310 ∪ cuni 3683 Ord word 4222 Oncon0 4223 Lim wlim 4224 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-uni 3684 df-tr 3967 df-iord 4226 df-on 4228 df-ilim 4229 df-suc 4231 |
This theorem is referenced by: (None) |
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