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Mirrors > Home > MPE Home > Th. List > ordsuc | Structured version Visualization version Unicode version |
Description: The successor of an ordinal class is ordinal. (Contributed by NM, 3-Apr-1995.) |
Ref | Expression |
---|---|
ordsuc |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elong 5431 |
. . . 4
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2 | suceloni 6640 |
. . . . 5
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3 | eloni 5433 |
. . . . 5
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4 | 2, 3 | syl 17 |
. . . 4
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5 | 1, 4 | syl6bir 233 |
. . 3
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6 | sucidg 5501 |
. . . 4
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7 | ordelord 5445 |
. . . . 5
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8 | 7 | ex 436 |
. . . 4
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9 | 6, 8 | syl5com 31 |
. . 3
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10 | 5, 9 | impbid 194 |
. 2
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11 | sucprc 5498 |
. . . 4
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12 | 11 | eqcomd 2457 |
. . 3
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13 | ordeq 5430 |
. . 3
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14 | 12, 13 | syl 17 |
. 2
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15 | 10, 14 | pm2.61i 168 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-sep 4525 ax-nul 4534 ax-pr 4639 ax-un 6583 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-ral 2742 df-rex 2743 df-rab 2746 df-v 3047 df-sbc 3268 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-pss 3420 df-nul 3732 df-if 3882 df-sn 3969 df-pr 3971 df-tp 3973 df-op 3975 df-uni 4199 df-br 4403 df-opab 4462 df-tr 4498 df-eprel 4745 df-po 4755 df-so 4756 df-fr 4793 df-we 4795 df-ord 5426 df-on 5427 df-suc 5429 |
This theorem is referenced by: ordpwsuc 6642 sucelon 6644 ordsucss 6645 onpsssuc 6646 ordsucelsuc 6649 ordsucsssuc 6650 ordsucuniel 6651 ordsucun 6652 onsucuni2 6661 0elsuc 6662 nlimsucg 6669 limsssuc 6677 php4 7759 cantnflt 8177 fin23lem26 8755 hsmexlem1 8856 onsuct0 31101 |
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