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Mirrors > Home > MPE Home > Th. List > alephfnon | Structured version Visualization version GIF version |
Description: The aleph function is a function on the class of ordinal numbers. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 13-Sep-2013.) |
Ref | Expression |
---|---|
alephfnon | ⊢ ℵ Fn On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdgfnon 8437 | . 2 ⊢ rec(har, ω) Fn On | |
2 | df-aleph 9963 | . . 3 ⊢ ℵ = rec(har, ω) | |
3 | 2 | fneq1i 6646 | . 2 ⊢ (ℵ Fn On ↔ rec(har, ω) Fn On) |
4 | 1, 3 | mpbir 230 | 1 ⊢ ℵ Fn On |
Colors of variables: wff setvar class |
Syntax hints: Oncon0 6364 Fn wfn 6538 ωcom 7868 reccrdg 8428 harchar 9579 ℵcale 9959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-aleph 9963 |
This theorem is referenced by: alephon 10092 alephcard 10093 alephnbtwn 10094 alephgeom 10105 alephf1 10108 infenaleph 10114 isinfcard 10115 alephiso 10121 alephsmo 10125 alephf1ALT 10126 alephfplem1 10127 alephfplem3 10129 alephsing 10299 alephadd 10600 alephreg 10605 pwcfsdom 10606 cfpwsdom 10607 gch2 10698 gch3 10699 |
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