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Mirrors > Home > MPE Home > Th. List > r1sucg | 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 Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
r1sucg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdgsucg 6644 |
. . 3
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2 | df-r1 7650 |
. . . 4
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3 | 2 | dmeqi 5034 |
. . 3
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4 | 1, 3 | eleq2s 2500 |
. 2
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5 | 2 | fveq1i 5692 |
. 2
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6 | fvex 5705 |
. . . 4
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7 | pweq 3766 |
. . . . 5
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8 | eqid 2408 |
. . . . 5
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9 | 6 | pwex 4346 |
. . . . 5
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10 | 7, 8, 9 | fvmpt 5769 |
. . . 4
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11 | 6, 10 | ax-mp 8 |
. . 3
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12 | 2 | fveq1i 5692 |
. . . 4
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13 | 12 | fveq2i 5694 |
. . 3
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14 | 11, 13 | eqtr3i 2430 |
. 2
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15 | 4, 5, 14 | 3eqtr4g 2465 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: r1suc 7656 r1fin 7659 r1tr 7662 r1ordg 7664 r1pwss 7670 r1val1 7672 rankwflemb 7679 r1elwf 7682 rankr1ai 7684 rankr1bg 7689 pwwf 7693 unwf 7696 uniwf 7705 rankonidlem 7714 rankr1id 7748 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2389 ax-sep 4294 ax-nul 4302 ax-pow 4341 ax-pr 4367 ax-un 4664 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2262 df-mo 2263 df-clab 2395 df-cleq 2401 df-clel 2404 df-nfc 2533 df-ne 2573 df-ral 2675 df-rex 2676 df-reu 2677 df-rab 2679 df-v 2922 df-sbc 3126 df-csb 3216 df-dif 3287 df-un 3289 df-in 3291 df-ss 3298 df-pss 3300 df-nul 3593 df-if 3704 df-pw 3765 df-sn 3784 df-pr 3785 df-tp 3786 df-op 3787 df-uni 3980 df-iun 4059 df-br 4177 df-opab 4231 df-mpt 4232 df-tr 4267 df-eprel 4458 df-id 4462 df-po 4467 df-so 4468 df-fr 4505 df-we 4507 df-ord 4548 df-on 4549 df-lim 4550 df-suc 4551 df-xp 4847 df-rel 4848 df-cnv 4849 df-co 4850 df-dm 4851 df-rn 4852 df-res 4853 df-ima 4854 df-iota 5381 df-fun 5419 df-fn 5420 df-f 5421 df-f1 5422 df-fo 5423 df-f1o 5424 df-fv 5425 df-recs 6596 df-rdg 6631 df-r1 7650 |
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