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Mirrors > Home > MPE Home > Th. List > bdayimaon | Structured version Visualization version GIF version |
Description: Lemma for full-eta properties. The successor of the union of the image of the birthday function under a set is an ordinal. (Contributed by Scott Fenton, 20-Aug-2011.) |
Ref | Expression |
---|---|
bdayimaon | β’ (π΄ β π β suc βͺ ( bday β π΄) β On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdayfo 27048 | . . . . . 6 β’ bday : No βontoβOn | |
2 | fofun 6761 | . . . . . 6 β’ ( bday : No βontoβOn β Fun bday ) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 β’ Fun bday |
4 | funimaexg 6591 | . . . . 5 β’ ((Fun bday β§ π΄ β π) β ( bday β π΄) β V) | |
5 | 3, 4 | mpan 689 | . . . 4 β’ (π΄ β π β ( bday β π΄) β V) |
6 | 5 | uniexd 7683 | . . 3 β’ (π΄ β π β βͺ ( bday β π΄) β V) |
7 | imassrn 6028 | . . . . 5 β’ ( bday β π΄) β ran bday | |
8 | forn 6763 | . . . . . 6 β’ ( bday : No βontoβOn β ran bday = On) | |
9 | 1, 8 | ax-mp 5 | . . . . 5 β’ ran bday = On |
10 | 7, 9 | sseqtri 3984 | . . . 4 β’ ( bday β π΄) β On |
11 | ssorduni 7717 | . . . 4 β’ (( bday β π΄) β On β Ord βͺ ( bday β π΄)) | |
12 | 10, 11 | ax-mp 5 | . . 3 β’ Ord βͺ ( bday β π΄) |
13 | 6, 12 | jctil 521 | . 2 β’ (π΄ β π β (Ord βͺ ( bday β π΄) β§ βͺ ( bday β π΄) β V)) |
14 | elon2 6332 | . . 3 β’ (βͺ ( bday β π΄) β On β (Ord βͺ ( bday β π΄) β§ βͺ ( bday β π΄) β V)) | |
15 | onsucb 7756 | . . 3 β’ (βͺ ( bday β π΄) β On β suc βͺ ( bday β π΄) β On) | |
16 | 14, 15 | bitr3i 277 | . 2 β’ ((Ord βͺ ( bday β π΄) β§ βͺ ( bday β π΄) β V) β suc βͺ ( bday β π΄) β On) |
17 | 13, 16 | sylib 217 | 1 β’ (π΄ β π β suc βͺ ( bday β π΄) β On) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3447 β wss 3914 βͺ cuni 4869 ran crn 5638 β cima 5640 Ord word 6320 Oncon0 6321 suc csuc 6323 Fun wfun 6494 βontoβwfo 6498 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-3or 1089 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-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 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-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 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-ord 6324 df-on 6325 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: noetasuplem1 27104 noetainflem1 27108 |
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