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Mirrors > Home > MPE Home > Th. List > r1suc | Structured version Visualization version Unicode version |
Description: Value of the cumulative hierarchy of sets function at a successor ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 2-Sep-2003.) (Revised by Mario Carneiro, 10-Sep-2013.) |
Ref | Expression |
---|---|
r1suc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r1sucg 8632 | . 2 | |
2 | r1fnon 8630 | . . . 4 | |
3 | fndm 5990 | . . . 4 | |
4 | 2, 3 | ax-mp 5 | . . 3 |
5 | 4 | eqcomi 2631 | . 2 |
6 | 1, 5 | eleq2s 2719 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 cpw 4158 cdm 5114 con0 5723 csuc 5725 wfn 5883 cfv 5888 cr1 8625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-r1 8627 |
This theorem is referenced by: r1sdom 8637 r1sssuc 8646 tz9.12lem3 8652 rankval2 8681 rankpwi 8686 dfac12lem2 8966 dfac12r 8968 ackbij2lem2 9062 ackbij2lem3 9063 wunr1om 9541 r1wunlim 9559 tskr1om 9589 inar1 9597 inatsk 9600 grur1a 9641 grothomex 9651 rankeq1o 32278 elhf2 32282 0hf 32284 aomclem1 37624 |
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