![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > onsuci | GIF version |
Description: The successor of an ordinal number is an ordinal number. Inference associated with onsuc 4502 and onsucb 4504. 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 | onsuc 4502 | . 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 2148 Oncon0 4365 suc csuc 4367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2741 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-uni 3812 df-tr 4104 df-iord 4368 df-on 4370 df-suc 4373 |
This theorem is referenced by: ordtri2orexmid 4524 onsucsssucexmid 4528 ordsoexmid 4563 ordtri2or2exmid 4572 ontri2orexmidim 4573 tfr0dm 6325 1on 6426 2on 6428 3on 6430 4on 6431 onntri35 7238 onntri45 7242 prarloclemarch2 7420 |
Copyright terms: Public domain | W3C validator |