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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 9416, 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 6863 | . 2 β’ (π β (harβπ) β π β V) | |
2 | reldom 8810 | . . . 4 β’ Rel βΌ | |
3 | 2 | brrelex2i 5675 | . . 3 β’ (π βΌ π β π β V) |
4 | 3 | adantl 482 | . 2 β’ ((π β On β§ π βΌ π) β π β V) |
5 | harval 9417 | . . . 4 β’ (π β V β (harβπ) = {π¦ β On β£ π¦ βΌ π}) | |
6 | 5 | eleq2d 2822 | . . 3 β’ (π β V β (π β (harβπ) β π β {π¦ β On β£ π¦ βΌ π})) |
7 | breq1 5095 | . . . 4 β’ (π¦ = π β (π¦ βΌ π β π βΌ π)) | |
8 | 7 | elrab 3634 | . . 3 β’ (π β {π¦ β On β£ π¦ βΌ π} β (π β On β§ π βΌ π)) |
9 | 6, 8 | bitrdi 286 | . 2 β’ (π β V β (π β (harβπ) β (π β On β§ π βΌ π))) |
10 | 1, 4, 9 | pm5.21nii 379 | 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 396 β wcel 2105 {crab 3403 Vcvv 3441 class class class wbr 5092 Oncon0 6302 βcfv 6479 βΌ cdom 8802 harchar 9413 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-se 5576 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-isom 6488 df-riota 7293 df-ov 7340 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-en 8805 df-dom 8806 df-oi 9367 df-har 9414 |
This theorem is referenced by: harndom 9419 harcard 9835 cardprclem 9836 cardaleph 9946 dfac12lem2 10001 hsmexlem1 10283 pwcfsdom 10440 pwfseqlem5 10520 hargch 10530 harinf 41119 harn0 41190 |
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