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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 8448 | . 2 ⊢ rec(har, ω) Fn On | |
2 | df-aleph 9983 | . . 3 ⊢ ℵ = rec(har, ω) | |
3 | 2 | fneq1i 6657 | . 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 6376 Fn wfn 6549 ωcom 7876 reccrdg 8439 harchar 9599 ℵcale 9979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-aleph 9983 |
This theorem is referenced by: alephon 10112 alephcard 10113 alephnbtwn 10114 alephgeom 10125 alephf1 10128 infenaleph 10134 isinfcard 10135 alephiso 10141 alephsmo 10145 alephf1ALT 10146 alephfplem1 10147 alephfplem3 10149 alephsing 10319 alephadd 10620 alephreg 10625 pwcfsdom 10626 cfpwsdom 10627 gch2 10718 gch3 10719 |
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