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Mirrors > Home > ILE Home > Th. List > onsuci | GIF version |
Description: The successor of an ordinal number is an ordinal number. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.) |
Ref | Expression |
---|---|
onssi.1 | ⊢ 𝐴 ∈ On |
Ref | Expression |
---|---|
onsuci | ⊢ suc 𝐴 ∈ On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onssi.1 | . 2 ⊢ 𝐴 ∈ On | |
2 | suceloni 4472 | . 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ suc 𝐴 ∈ On |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2135 Oncon0 4335 suc csuc 4337 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-uni 3784 df-tr 4075 df-iord 4338 df-on 4340 df-suc 4343 |
This theorem is referenced by: ordtri2orexmid 4494 onsucsssucexmid 4498 ordsoexmid 4533 ordtri2or2exmid 4542 ontri2orexmidim 4543 tfr0dm 6281 1on 6382 2on 6384 3on 6386 4on 6387 onntri35 7184 onntri45 7188 prarloclemarch2 7351 |
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