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Mirrors > Home > MPE Home > Th. List > cardsdomel | Structured version Visualization version GIF version |
Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 15-Jan-2013.) (Revised by Mario Carneiro, 4-Jun-2015.) |
Ref | Expression |
---|---|
cardsdomel | β’ ((π΄ β On β§ π΅ β dom card) β (π΄ βΊ π΅ β π΄ β (cardβπ΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardid2 9976 | . . . . . . 7 β’ (π΅ β dom card β (cardβπ΅) β π΅) | |
2 | 1 | ensymd 9024 | . . . . . 6 β’ (π΅ β dom card β π΅ β (cardβπ΅)) |
3 | sdomentr 9134 | . . . . . 6 β’ ((π΄ βΊ π΅ β§ π΅ β (cardβπ΅)) β π΄ βΊ (cardβπ΅)) | |
4 | 2, 3 | sylan2 591 | . . . . 5 β’ ((π΄ βΊ π΅ β§ π΅ β dom card) β π΄ βΊ (cardβπ΅)) |
5 | ssdomg 9019 | . . . . . . . 8 β’ (π΄ β On β ((cardβπ΅) β π΄ β (cardβπ΅) βΌ π΄)) | |
6 | cardon 9967 | . . . . . . . . 9 β’ (cardβπ΅) β On | |
7 | domtriord 9146 | . . . . . . . . 9 β’ (((cardβπ΅) β On β§ π΄ β On) β ((cardβπ΅) βΌ π΄ β Β¬ π΄ βΊ (cardβπ΅))) | |
8 | 6, 7 | mpan 688 | . . . . . . . 8 β’ (π΄ β On β ((cardβπ΅) βΌ π΄ β Β¬ π΄ βΊ (cardβπ΅))) |
9 | 5, 8 | sylibd 238 | . . . . . . 7 β’ (π΄ β On β ((cardβπ΅) β π΄ β Β¬ π΄ βΊ (cardβπ΅))) |
10 | 9 | con2d 134 | . . . . . 6 β’ (π΄ β On β (π΄ βΊ (cardβπ΅) β Β¬ (cardβπ΅) β π΄)) |
11 | ontri1 6398 | . . . . . . . 8 β’ (((cardβπ΅) β On β§ π΄ β On) β ((cardβπ΅) β π΄ β Β¬ π΄ β (cardβπ΅))) | |
12 | 6, 11 | mpan 688 | . . . . . . 7 β’ (π΄ β On β ((cardβπ΅) β π΄ β Β¬ π΄ β (cardβπ΅))) |
13 | 12 | con2bid 353 | . . . . . 6 β’ (π΄ β On β (π΄ β (cardβπ΅) β Β¬ (cardβπ΅) β π΄)) |
14 | 10, 13 | sylibrd 258 | . . . . 5 β’ (π΄ β On β (π΄ βΊ (cardβπ΅) β π΄ β (cardβπ΅))) |
15 | 4, 14 | syl5 34 | . . . 4 β’ (π΄ β On β ((π΄ βΊ π΅ β§ π΅ β dom card) β π΄ β (cardβπ΅))) |
16 | 15 | expcomd 415 | . . 3 β’ (π΄ β On β (π΅ β dom card β (π΄ βΊ π΅ β π΄ β (cardβπ΅)))) |
17 | 16 | imp 405 | . 2 β’ ((π΄ β On β§ π΅ β dom card) β (π΄ βΊ π΅ β π΄ β (cardβπ΅))) |
18 | cardsdomelir 9996 | . 2 β’ (π΄ β (cardβπ΅) β π΄ βΊ π΅) | |
19 | 17, 18 | impbid1 224 | 1 β’ ((π΄ β On β§ π΅ β dom card) β (π΄ βΊ π΅ β π΄ β (cardβπ΅))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 394 β wcel 2098 β wss 3939 class class class wbr 5143 dom cdm 5672 Oncon0 6364 βcfv 6543 β cen 8959 βΌ cdom 8960 βΊ csdm 8961 cardccrd 9958 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6367 df-on 6368 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-card 9962 |
This theorem is referenced by: iscard 9998 cardval2 10014 infxpenlem 10036 alephnbtwn 10094 alephnbtwn2 10095 alephord2 10099 alephsdom 10109 pwsdompw 10227 inaprc 10859 |
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