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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > harinf | Structured version Visualization version GIF version |
Description: The Hartogs number of an infinite set is at least Ο. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) |
Ref | Expression |
---|---|
harinf | β’ ((π β π β§ Β¬ π β Fin) β Ο β (harβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnon 7857 | . . . . 5 β’ (π₯ β Ο β π₯ β On) | |
2 | 1 | adantl 481 | . . . 4 β’ (((π β π β§ Β¬ π β Fin) β§ π₯ β Ο) β π₯ β On) |
3 | simplr 766 | . . . . . 6 β’ (((π β π β§ Β¬ π β Fin) β§ π₯ β Ο) β Β¬ π β Fin) | |
4 | nnfi 9166 | . . . . . . . 8 β’ (π₯ β Ο β π₯ β Fin) | |
5 | 4 | adantl 481 | . . . . . . 7 β’ (((π β π β§ Β¬ π β Fin) β§ π₯ β Ο) β π₯ β Fin) |
6 | sdomdom 8975 | . . . . . . 7 β’ (π βΊ π₯ β π βΌ π₯) | |
7 | domfi 9191 | . . . . . . . 8 β’ ((π₯ β Fin β§ π βΌ π₯) β π β Fin) | |
8 | 7 | ex 412 | . . . . . . 7 β’ (π₯ β Fin β (π βΌ π₯ β π β Fin)) |
9 | 5, 6, 8 | syl2im 40 | . . . . . 6 β’ (((π β π β§ Β¬ π β Fin) β§ π₯ β Ο) β (π βΊ π₯ β π β Fin)) |
10 | 3, 9 | mtod 197 | . . . . 5 β’ (((π β π β§ Β¬ π β Fin) β§ π₯ β Ο) β Β¬ π βΊ π₯) |
11 | simpll 764 | . . . . . 6 β’ (((π β π β§ Β¬ π β Fin) β§ π₯ β Ο) β π β π) | |
12 | fidomtri 9987 | . . . . . 6 β’ ((π₯ β Fin β§ π β π) β (π₯ βΌ π β Β¬ π βΊ π₯)) | |
13 | 5, 11, 12 | syl2anc 583 | . . . . 5 β’ (((π β π β§ Β¬ π β Fin) β§ π₯ β Ο) β (π₯ βΌ π β Β¬ π βΊ π₯)) |
14 | 10, 13 | mpbird 257 | . . . 4 β’ (((π β π β§ Β¬ π β Fin) β§ π₯ β Ο) β π₯ βΌ π) |
15 | elharval 9555 | . . . 4 β’ (π₯ β (harβπ) β (π₯ β On β§ π₯ βΌ π)) | |
16 | 2, 14, 15 | sylanbrc 582 | . . 3 β’ (((π β π β§ Β¬ π β Fin) β§ π₯ β Ο) β π₯ β (harβπ)) |
17 | 16 | ex 412 | . 2 β’ ((π β π β§ Β¬ π β Fin) β (π₯ β Ο β π₯ β (harβπ))) |
18 | 17 | ssrdv 3983 | 1 β’ ((π β π β§ Β¬ π β Fin) β Ο β (harβπ)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 395 β wcel 2098 β wss 3943 class class class wbr 5141 Oncon0 6357 βcfv 6536 Οcom 7851 βΌ cdom 8936 βΊ csdm 8937 Fincfn 8938 harchar 9550 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-1o 8464 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-oi 9504 df-har 9551 df-card 9933 |
This theorem is referenced by: ttac 42334 |
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