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Mirrors > Home > MPE Home > Th. List > cardsdomelir | Structured version Visualization version GIF version |
Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 9995 and can be proven without the AC. (Contributed by Mario Carneiro, 15-Jan-2013.) |
Ref | Expression |
---|---|
cardsdomelir | β’ (π΄ β (cardβπ΅) β π΄ βΊ π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardon 9965 | . . . 4 β’ (cardβπ΅) β On | |
2 | 1 | onelssi 6479 | . . . 4 β’ (π΄ β (cardβπ΅) β π΄ β (cardβπ΅)) |
3 | ssdomg 9017 | . . . 4 β’ ((cardβπ΅) β On β (π΄ β (cardβπ΅) β π΄ βΌ (cardβπ΅))) | |
4 | 1, 2, 3 | mpsyl 68 | . . 3 β’ (π΄ β (cardβπ΅) β π΄ βΌ (cardβπ΅)) |
5 | elfvdm 6928 | . . . 4 β’ (π΄ β (cardβπ΅) β π΅ β dom card) | |
6 | cardid2 9974 | . . . 4 β’ (π΅ β dom card β (cardβπ΅) β π΅) | |
7 | 5, 6 | syl 17 | . . 3 β’ (π΄ β (cardβπ΅) β (cardβπ΅) β π΅) |
8 | domentr 9030 | . . 3 β’ ((π΄ βΌ (cardβπ΅) β§ (cardβπ΅) β π΅) β π΄ βΌ π΅) | |
9 | 4, 7, 8 | syl2anc 582 | . 2 β’ (π΄ β (cardβπ΅) β π΄ βΌ π΅) |
10 | cardne 9986 | . 2 β’ (π΄ β (cardβπ΅) β Β¬ π΄ β π΅) | |
11 | brsdom 8992 | . 2 β’ (π΄ βΊ π΅ β (π΄ βΌ π΅ β§ Β¬ π΄ β π΅)) | |
12 | 9, 10, 11 | sylanbrc 581 | 1 β’ (π΄ β (cardβπ΅) β π΄ βΊ π΅) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wcel 2098 β wss 3940 class class class wbr 5143 dom cdm 5672 Oncon0 6364 βcfv 6542 β cen 8957 βΌ cdom 8958 βΊ csdm 8959 cardccrd 9956 |
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 7737 |
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 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 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 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-en 8961 df-dom 8962 df-sdom 8963 df-card 9960 |
This theorem is referenced by: cardsdomel 9995 pwsdompw 10225 alephval2 10593 pwcfsdom 10604 tskcard 10802 |
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