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| Description: The successor of an ordinal number is an ordinal number. Closed form of onsuci 4565. Forward implication of onsucb 4552. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.) |
| Ref | Expression |
|---|---|
| onsuc |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eloni 4423 |
. . 3
| |
| 2 | ordsucim 4549 |
. . 3
| |
| 3 | 1, 2 | syl 14 |
. 2
|
| 4 | sucexg 4547 |
. . 3
| |
| 5 | elong 4421 |
. . 3
| |
| 6 | 4, 5 | syl 14 |
. 2
|
| 7 | 3, 6 | mpbird 167 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| 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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4163 ax-pow 4219 ax-pr 4254 ax-un 4481 |
| This theorem depends on definitions: df-bi 117 df-tru 1376 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ral 2489 df-rex 2490 df-v 2774 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-uni 3851 df-tr 4144 df-iord 4414 df-on 4416 df-suc 4419 |
| This theorem is referenced by: onsucb 4552 unon 4560 onsuci 4565 ordsucunielexmid 4580 tfrlemisucaccv 6413 tfrexlem 6422 tfri1dALT 6439 rdgisuc1 6472 rdgon 6474 oacl 6548 oasuc 6552 omsuc 6560 |
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