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Mirrors > Home > MPE Home > Th. List > bdayfo | Structured version Visualization version GIF version |
Description: The birthday function maps the surreals onto the ordinals. Axiom B of [Alling] p. 184. (Proof shortened on 14-Apr-2012 by SF). (Contributed by Scott Fenton, 11-Jun-2011.) |
Ref | Expression |
---|---|
bdayfo | β’ bday : No βontoβOn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmexg 7844 | . . . 4 β’ (π₯ β No β dom π₯ β V) | |
2 | 1 | rgen 3063 | . . 3 β’ βπ₯ β No dom π₯ β V |
3 | df-bday 27016 | . . . 4 β’ bday = (π₯ β No β¦ dom π₯) | |
4 | 3 | mptfng 6644 | . . 3 β’ (βπ₯ β No dom π₯ β V β bday Fn No ) |
5 | 2, 4 | mpbi 229 | . 2 β’ bday Fn No |
6 | 3 | rnmpt 5914 | . . 3 β’ ran bday = {π¦ β£ βπ₯ β No π¦ = dom π₯} |
7 | noxp1o 27034 | . . . . . 6 β’ (π¦ β On β (π¦ Γ {1o}) β No ) | |
8 | 1oex 8426 | . . . . . . . . 9 β’ 1o β V | |
9 | 8 | snnz 4741 | . . . . . . . 8 β’ {1o} β β |
10 | dmxp 5888 | . . . . . . . 8 β’ ({1o} β β β dom (π¦ Γ {1o}) = π¦) | |
11 | 9, 10 | ax-mp 5 | . . . . . . 7 β’ dom (π¦ Γ {1o}) = π¦ |
12 | 11 | eqcomi 2742 | . . . . . 6 β’ π¦ = dom (π¦ Γ {1o}) |
13 | dmeq 5863 | . . . . . . 7 β’ (π₯ = (π¦ Γ {1o}) β dom π₯ = dom (π¦ Γ {1o})) | |
14 | 13 | rspceeqv 3599 | . . . . . 6 β’ (((π¦ Γ {1o}) β No β§ π¦ = dom (π¦ Γ {1o})) β βπ₯ β No π¦ = dom π₯) |
15 | 7, 12, 14 | sylancl 587 | . . . . 5 β’ (π¦ β On β βπ₯ β No π¦ = dom π₯) |
16 | nodmon 27021 | . . . . . . 7 β’ (π₯ β No β dom π₯ β On) | |
17 | eleq1a 2829 | . . . . . . 7 β’ (dom π₯ β On β (π¦ = dom π₯ β π¦ β On)) | |
18 | 16, 17 | syl 17 | . . . . . 6 β’ (π₯ β No β (π¦ = dom π₯ β π¦ β On)) |
19 | 18 | rexlimiv 3142 | . . . . 5 β’ (βπ₯ β No π¦ = dom π₯ β π¦ β On) |
20 | 15, 19 | impbii 208 | . . . 4 β’ (π¦ β On β βπ₯ β No π¦ = dom π₯) |
21 | 20 | abbi2i 2870 | . . 3 β’ On = {π¦ β£ βπ₯ β No π¦ = dom π₯} |
22 | 6, 21 | eqtr4i 2764 | . 2 β’ ran bday = On |
23 | df-fo 6506 | . 2 β’ ( bday : No βontoβOn β ( bday Fn No β§ ran bday = On)) | |
24 | 5, 22, 23 | mpbir2an 710 | 1 β’ bday : No βontoβOn |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 {cab 2710 β wne 2940 βwral 3061 βwrex 3070 Vcvv 3447 β c0 4286 {csn 4590 Γ cxp 5635 dom cdm 5637 ran crn 5638 Oncon0 6321 Fn wfn 6495 βontoβwfo 6498 1oc1o 8409 No csur 27011 bday cbday 27013 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-1o 8416 df-no 27014 df-bday 27016 |
This theorem is referenced by: nodense 27063 bdayimaon 27064 nosupno 27074 nosupbday 27076 noinfno 27089 noinfbday 27091 noetasuplem4 27107 noetainflem4 27111 bdayfun 27141 bdayfn 27142 bdaydm 27143 bdayrn 27144 bdayelon 27145 noprc 27148 noeta2 27153 |
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