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Mirrors > Home > ILE Home > Th. List > suceloni | Unicode version |
Description: The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.) |
Ref | Expression |
---|---|
suceloni |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 4358 | . . 3 | |
2 | ordsucim 4482 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | sucexg 4480 | . . 3 | |
5 | elong 4356 | . . 3 | |
6 | 4, 5 | syl 14 | . 2 |
7 | 3, 6 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wcel 2141 cvv 2730 word 4345 con0 4346 csuc 4348 |
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 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 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 |
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-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-uni 3795 df-tr 4086 df-iord 4349 df-on 4351 df-suc 4354 |
This theorem is referenced by: sucelon 4485 unon 4493 onsuci 4498 ordsucunielexmid 4513 tfrlemisucaccv 6301 tfrexlem 6310 tfri1dALT 6327 rdgisuc1 6360 rdgon 6362 oacl 6436 oasuc 6440 omsuc 6448 |
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