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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 6281 | . 2 ⊢ 1o = suc ∅ | |
2 | 0elon 4284 | . . 3 ⊢ ∅ ∈ On | |
3 | 2 | onsuci 4402 | . 2 ⊢ suc ∅ ∈ On |
4 | 1, 3 | eqeltri 2190 | 1 ⊢ 1o ∈ On |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1465 ∅c0 3333 Oncon0 4255 suc csuc 4257 1oc1o 6274 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-uni 3707 df-tr 3997 df-iord 4258 df-on 4260 df-suc 4263 df-1o 6281 |
This theorem is referenced by: 1oex 6289 2on 6290 2on0 6291 2oconcl 6304 fnoei 6316 oeiexg 6317 oeiv 6320 oei0 6323 oeicl 6326 o1p1e2 6332 oawordriexmid 6334 enpr2d 6679 endisj 6686 snexxph 6806 djuex 6896 1stinr 6929 2ndinr 6930 pm54.43 7014 xpdjuen 7042 prarloclemarch2 7195 bj-el2oss1o 12908 nnsf 13126 |
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