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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 6395 | . 2 ⊢ 1o = suc ∅ | |
2 | 0elon 4377 | . . 3 ⊢ ∅ ∈ On | |
3 | 2 | onsuci 4500 | . 2 ⊢ suc ∅ ∈ On |
4 | 1, 3 | eqeltri 2243 | 1 ⊢ 1o ∈ On |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2141 ∅c0 3414 Oncon0 4348 suc csuc 4350 1oc1o 6388 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-uni 3797 df-tr 4088 df-iord 4351 df-on 4353 df-suc 4356 df-1o 6395 |
This theorem is referenced by: 1oex 6403 2on 6404 2on0 6405 2oconcl 6418 fnoei 6431 oeiexg 6432 oeiv 6435 oei0 6438 oeicl 6441 o1p1e2 6447 oawordriexmid 6449 enpr2d 6795 endisj 6802 snexxph 6927 djuex 7020 1stinr 7053 2ndinr 7054 pm54.43 7167 xpdjuen 7195 prarloclemarch2 7381 bj-el2oss1o 13809 nnsf 14038 |
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