![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7858 | . 2 ⊢ rec(har, ω) Fn On | |
2 | df-aleph 9163 | . . 3 ⊢ ℵ = rec(har, ω) | |
3 | 2 | fneq1i 6283 | . 2 ⊢ (ℵ Fn On ↔ rec(har, ω) Fn On) |
4 | 1, 3 | mpbir 223 | 1 ⊢ ℵ Fn On |
Colors of variables: wff setvar class |
Syntax hints: Oncon0 6029 Fn wfn 6183 ωcom 7396 reccrdg 7849 harchar 8815 ℵcale 9159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-aleph 9163 |
This theorem is referenced by: alephon 9289 alephcard 9290 alephnbtwn 9291 alephgeom 9302 alephf1 9305 infenaleph 9311 isinfcard 9312 alephiso 9318 alephsmo 9322 alephf1ALT 9323 alephfplem1 9324 alephfplem3 9326 alephsing 9496 alephadd 9797 alephreg 9802 pwcfsdom 9803 cfpwsdom 9804 gch2 9895 gch3 9896 |
Copyright terms: Public domain | W3C validator |