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Mirrors > Home > ILE Home > Th. List > 1on | GIF version |
Description: Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.) |
Ref | Expression |
---|---|
1on | ⊢ 1o ∈ On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-1o 6306 | . 2 ⊢ 1o = suc ∅ | |
2 | 0elon 4309 | . . 3 ⊢ ∅ ∈ On | |
3 | 2 | onsuci 4427 | . 2 ⊢ suc ∅ ∈ On |
4 | 1, 3 | eqeltri 2210 | 1 ⊢ 1o ∈ On |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 ∅c0 3358 Oncon0 4280 suc csuc 4282 1oc1o 6299 |
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-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-uni 3732 df-tr 4022 df-iord 4283 df-on 4285 df-suc 4288 df-1o 6306 |
This theorem is referenced by: 1oex 6314 2on 6315 2on0 6316 2oconcl 6329 fnoei 6341 oeiexg 6342 oeiv 6345 oei0 6348 oeicl 6351 o1p1e2 6357 oawordriexmid 6359 enpr2d 6704 endisj 6711 snexxph 6831 djuex 6921 1stinr 6954 2ndinr 6955 pm54.43 7039 xpdjuen 7067 prarloclemarch2 7220 bj-el2oss1o 12970 nnsf 13188 |
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