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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 9983 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 9953 | . . . 4 β’ (cardβπ΅) β On | |
2 | 1 | onelssi 6478 | . . . 4 β’ (π΄ β (cardβπ΅) β π΄ β (cardβπ΅)) |
3 | ssdomg 9010 | . . . 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 9962 | . . . 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 9023 | . . 3 β’ ((π΄ βΌ (cardβπ΅) β§ (cardβπ΅) β π΅) β π΄ βΌ π΅) | |
9 | 4, 7, 8 | syl2anc 583 | . 2 β’ (π΄ β (cardβπ΅) β π΄ βΌ π΅) |
10 | cardne 9974 | . 2 β’ (π΄ β (cardβπ΅) β Β¬ π΄ β π΅) | |
11 | brsdom 8985 | . 2 β’ (π΄ βΊ π΅ β (π΄ βΌ π΅ β§ Β¬ π΄ β π΅)) | |
12 | 9, 10, 11 | sylanbrc 582 | 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 2099 β wss 3944 class class class wbr 5142 dom cdm 5672 Oncon0 6363 βcfv 6542 β cen 8950 βΌ cdom 8951 βΊ csdm 8952 cardccrd 9944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 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 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6366 df-on 6367 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 8954 df-dom 8955 df-sdom 8956 df-card 9948 |
This theorem is referenced by: cardsdomel 9983 pwsdompw 10213 alephval2 10581 pwcfsdom 10592 tskcard 10790 |
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