![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7888 | . . . 4 β’ (π₯ β No β dom π₯ β V) | |
2 | 1 | rgen 3055 | . . 3 β’ βπ₯ β No dom π₯ β V |
3 | df-bday 27519 | . . . 4 β’ bday = (π₯ β No β¦ dom π₯) | |
4 | 3 | mptfng 6680 | . . 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 5945 | . . 3 β’ ran bday = {π¦ β£ βπ₯ β No π¦ = dom π₯} |
7 | noxp1o 27537 | . . . . . 6 β’ (π¦ β On β (π¦ Γ {1o}) β No ) | |
8 | 1oex 8472 | . . . . . . . . 9 β’ 1o β V | |
9 | 8 | snnz 4773 | . . . . . . . 8 β’ {1o} β β |
10 | dmxp 5919 | . . . . . . . 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 2733 | . . . . . 6 β’ π¦ = dom (π¦ Γ {1o}) |
13 | dmeq 5894 | . . . . . . 7 β’ (π₯ = (π¦ Γ {1o}) β dom π₯ = dom (π¦ Γ {1o})) | |
14 | 13 | rspceeqv 3626 | . . . . . 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 585 | . . . . 5 β’ (π¦ β On β βπ₯ β No π¦ = dom π₯) |
16 | nodmon 27524 | . . . . . . 7 β’ (π₯ β No β dom π₯ β On) | |
17 | eleq1a 2820 | . . . . . . 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 3140 | . . . . 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 | eqabi 2861 | . . 3 β’ On = {π¦ β£ βπ₯ β No π¦ = dom π₯} |
22 | 6, 21 | eqtr4i 2755 | . 2 β’ ran bday = On |
23 | df-fo 6540 | . 2 β’ ( bday : No βontoβOn β ( bday Fn No β§ ran bday = On)) | |
24 | 5, 22, 23 | mpbir2an 708 | 1 β’ bday : No βontoβOn |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 {cab 2701 β wne 2932 βwral 3053 βwrex 3062 Vcvv 3466 β c0 4315 {csn 4621 Γ cxp 5665 dom cdm 5667 ran crn 5668 Oncon0 6355 Fn wfn 6529 βontoβwfo 6532 1oc1o 8455 No csur 27514 bday cbday 27516 |
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 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-1o 8462 df-no 27517 df-bday 27519 |
This theorem is referenced by: nodense 27566 bdayimaon 27567 nosupno 27577 nosupbday 27579 noinfno 27592 noinfbday 27594 noetasuplem4 27610 noetainflem4 27614 bdayfun 27646 bdayfn 27647 bdaydm 27648 bdayrn 27649 bdayelon 27650 noprc 27653 noeta2 27658 |
Copyright terms: Public domain | W3C validator |