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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 7514 | . 2 ⊢ rec(har, ω) Fn On | |
2 | df-aleph 8766 | . . 3 ⊢ ℵ = rec(har, ω) | |
3 | 2 | fneq1i 5985 | . 2 ⊢ (ℵ Fn On ↔ rec(har, ω) Fn On) |
4 | 1, 3 | mpbir 221 | 1 ⊢ ℵ Fn On |
Colors of variables: wff setvar class |
Syntax hints: Oncon0 5723 Fn wfn 5883 ωcom 7065 reccrdg 7505 harchar 8461 ℵcale 8762 |
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-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-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-aleph 8766 |
This theorem is referenced by: alephon 8892 alephcard 8893 alephnbtwn 8894 alephgeom 8905 alephf1 8908 infenaleph 8914 isinfcard 8915 alephiso 8921 alephsmo 8925 alephf1ALT 8926 alephfplem1 8927 alephfplem3 8929 alephsing 9098 alephadd 9399 alephreg 9404 pwcfsdom 9405 cfpwsdom 9406 gch2 9497 gch3 9498 |
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