![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 9915 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 9885 | . . . 4 β’ (cardβπ΅) β On | |
2 | 1 | onelssi 6433 | . . . 4 β’ (π΄ β (cardβπ΅) β π΄ β (cardβπ΅)) |
3 | ssdomg 8943 | . . . 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 6880 | . . . 4 β’ (π΄ β (cardβπ΅) β π΅ β dom card) | |
6 | cardid2 9894 | . . . 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 8956 | . . 3 β’ ((π΄ βΌ (cardβπ΅) β§ (cardβπ΅) β π΅) β π΄ βΌ π΅) | |
9 | 4, 7, 8 | syl2anc 585 | . 2 β’ (π΄ β (cardβπ΅) β π΄ βΌ π΅) |
10 | cardne 9906 | . 2 β’ (π΄ β (cardβπ΅) β Β¬ π΄ β π΅) | |
11 | brsdom 8918 | . 2 β’ (π΄ βΊ π΅ β (π΄ βΌ π΅ β§ Β¬ π΄ β π΅)) | |
12 | 9, 10, 11 | sylanbrc 584 | 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 2107 β wss 3911 class class class wbr 5106 dom cdm 5634 Oncon0 6318 βcfv 6497 β cen 8883 βΌ cdom 8884 βΊ csdm 8885 cardccrd 9876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-en 8887 df-dom 8888 df-sdom 8889 df-card 9880 |
This theorem is referenced by: cardsdomel 9915 pwsdompw 10145 alephval2 10513 pwcfsdom 10524 tskcard 10722 |
Copyright terms: Public domain | W3C validator |