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Mirrors > Home > MPE Home > Th. List > elharval | Structured version Visualization version GIF version |
Description: The Hartogs number of a set contains exactly the ordinals that set dominates. Combined with harcl 9554, this implies that the Hartogs number of a set is greater than all ordinals that set dominates. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
elharval | β’ (π β (harβπ) β (π β On β§ π βΌ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvex 6930 | . 2 β’ (π β (harβπ) β π β V) | |
2 | reldom 8945 | . . . 4 β’ Rel βΌ | |
3 | 2 | brrelex2i 5734 | . . 3 β’ (π βΌ π β π β V) |
4 | 3 | adantl 483 | . 2 β’ ((π β On β§ π βΌ π) β π β V) |
5 | harval 9555 | . . . 4 β’ (π β V β (harβπ) = {π¦ β On β£ π¦ βΌ π}) | |
6 | 5 | eleq2d 2820 | . . 3 β’ (π β V β (π β (harβπ) β π β {π¦ β On β£ π¦ βΌ π})) |
7 | breq1 5152 | . . . 4 β’ (π¦ = π β (π¦ βΌ π β π βΌ π)) | |
8 | 7 | elrab 3684 | . . 3 β’ (π β {π¦ β On β£ π¦ βΌ π} β (π β On β§ π βΌ π)) |
9 | 6, 8 | bitrdi 287 | . 2 β’ (π β V β (π β (harβπ) β (π β On β§ π βΌ π))) |
10 | 1, 4, 9 | pm5.21nii 380 | 1 β’ (π β (harβπ) β (π β On β§ π βΌ π)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 397 β wcel 2107 {crab 3433 Vcvv 3475 class class class wbr 5149 Oncon0 6365 βcfv 6544 βΌ cdom 8937 harchar 9551 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
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 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-en 8940 df-dom 8941 df-oi 9505 df-har 9552 |
This theorem is referenced by: harndom 9557 harcard 9973 cardprclem 9974 cardaleph 10084 dfac12lem2 10139 hsmexlem1 10421 pwcfsdom 10578 pwfseqlem5 10658 hargch 10668 harinf 41773 harn0 41844 |
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