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Mirrors > Home > MPE Home > Th. List > cardsdom2 | Structured version Visualization version GIF version |
Description: A numerable set is strictly dominated by another iff their cardinalities are strictly ordered. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 29-Apr-2015.) |
Ref | Expression |
---|---|
cardsdom2 | β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΄) β (cardβπ΅) β π΄ βΊ π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | carddom2 9834 | . . 3 β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΄) β (cardβπ΅) β π΄ βΌ π΅)) | |
2 | carden2 9844 | . . . 4 β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΄) = (cardβπ΅) β π΄ β π΅)) | |
3 | 2 | necon3abid 2977 | . . 3 β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΄) β (cardβπ΅) β Β¬ π΄ β π΅)) |
4 | 1, 3 | anbi12d 631 | . 2 β’ ((π΄ β dom card β§ π΅ β dom card) β (((cardβπ΄) β (cardβπ΅) β§ (cardβπ΄) β (cardβπ΅)) β (π΄ βΌ π΅ β§ Β¬ π΄ β π΅))) |
5 | cardon 9801 | . . 3 β’ (cardβπ΄) β On | |
6 | cardon 9801 | . . 3 β’ (cardβπ΅) β On | |
7 | onelpss 6342 | . . 3 β’ (((cardβπ΄) β On β§ (cardβπ΅) β On) β ((cardβπ΄) β (cardβπ΅) β ((cardβπ΄) β (cardβπ΅) β§ (cardβπ΄) β (cardβπ΅)))) | |
8 | 5, 6, 7 | mp2an 689 | . 2 β’ ((cardβπ΄) β (cardβπ΅) β ((cardβπ΄) β (cardβπ΅) β§ (cardβπ΄) β (cardβπ΅))) |
9 | brsdom 8836 | . 2 β’ (π΄ βΊ π΅ β (π΄ βΌ π΅ β§ Β¬ π΄ β π΅)) | |
10 | 4, 8, 9 | 3bitr4g 313 | 1 β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΄) β (cardβπ΅) β π΄ βΊ π΅)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 396 β wcel 2105 β wne 2940 β wss 3898 class class class wbr 5092 dom cdm 5620 Oncon0 6302 βcfv 6479 β cen 8801 βΌ cdom 8802 βΊ csdm 8803 cardccrd 9792 |
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-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-rab 3404 df-v 3443 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-int 4895 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-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-ord 6305 df-on 6306 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-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-card 9796 |
This theorem is referenced by: domtri2 9846 nnsdomel 9847 indcardi 9898 sdom2en01 10159 cardsdom 10412 smobeth 10443 hargch 10530 cardpred 33361 |
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