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Mirrors > Home > MPE Home > Th. List > carddom2 | Structured version Visualization version GIF version |
Description: Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 10577, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
Ref | Expression |
---|---|
carddom2 | β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΄) β (cardβπ΅) β π΄ βΌ π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | carddomi2 9993 | . 2 β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΄) β (cardβπ΅) β π΄ βΌ π΅)) | |
2 | brdom2 9001 | . . 3 β’ (π΄ βΌ π΅ β (π΄ βΊ π΅ β¨ π΄ β π΅)) | |
3 | cardon 9967 | . . . . . . . 8 β’ (cardβπ΄) β On | |
4 | 3 | onelssi 6479 | . . . . . . 7 β’ ((cardβπ΅) β (cardβπ΄) β (cardβπ΅) β (cardβπ΄)) |
5 | carddomi2 9993 | . . . . . . . 8 β’ ((π΅ β dom card β§ π΄ β dom card) β ((cardβπ΅) β (cardβπ΄) β π΅ βΌ π΄)) | |
6 | 5 | ancoms 457 | . . . . . . 7 β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΅) β (cardβπ΄) β π΅ βΌ π΄)) |
7 | domnsym 9122 | . . . . . . 7 β’ (π΅ βΌ π΄ β Β¬ π΄ βΊ π΅) | |
8 | 4, 6, 7 | syl56 36 | . . . . . 6 β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΅) β (cardβπ΄) β Β¬ π΄ βΊ π΅)) |
9 | 8 | con2d 134 | . . . . 5 β’ ((π΄ β dom card β§ π΅ β dom card) β (π΄ βΊ π΅ β Β¬ (cardβπ΅) β (cardβπ΄))) |
10 | cardon 9967 | . . . . . 6 β’ (cardβπ΅) β On | |
11 | ontri1 6398 | . . . . . 6 β’ (((cardβπ΄) β On β§ (cardβπ΅) β On) β ((cardβπ΄) β (cardβπ΅) β Β¬ (cardβπ΅) β (cardβπ΄))) | |
12 | 3, 10, 11 | mp2an 690 | . . . . 5 β’ ((cardβπ΄) β (cardβπ΅) β Β¬ (cardβπ΅) β (cardβπ΄)) |
13 | 9, 12 | imbitrrdi 251 | . . . 4 β’ ((π΄ β dom card β§ π΅ β dom card) β (π΄ βΊ π΅ β (cardβπ΄) β (cardβπ΅))) |
14 | carden2b 9990 | . . . . . 6 β’ (π΄ β π΅ β (cardβπ΄) = (cardβπ΅)) | |
15 | eqimss 4031 | . . . . . 6 β’ ((cardβπ΄) = (cardβπ΅) β (cardβπ΄) β (cardβπ΅)) | |
16 | 14, 15 | syl 17 | . . . . 5 β’ (π΄ β π΅ β (cardβπ΄) β (cardβπ΅)) |
17 | 16 | a1i 11 | . . . 4 β’ ((π΄ β dom card β§ π΅ β dom card) β (π΄ β π΅ β (cardβπ΄) β (cardβπ΅))) |
18 | 13, 17 | jaod 857 | . . 3 β’ ((π΄ β dom card β§ π΅ β dom card) β ((π΄ βΊ π΅ β¨ π΄ β π΅) β (cardβπ΄) β (cardβπ΅))) |
19 | 2, 18 | biimtrid 241 | . 2 β’ ((π΄ β dom card β§ π΅ β dom card) β (π΄ βΌ π΅ β (cardβπ΄) β (cardβπ΅))) |
20 | 1, 19 | impbid 211 | 1 β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΄) β (cardβπ΅) β π΄ βΌ π΅)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 394 β¨ wo 845 = wceq 1533 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: carduni 10004 carden2 10010 cardsdom2 10011 domtri2 10012 infxpidm2 10040 cardaleph 10112 infenaleph 10114 alephinit 10118 ficardun2 10225 ficardun2OLD 10226 ackbij2 10266 cfflb 10282 fin1a2lem9 10431 carddom 10577 pwfseqlem5 10686 hashdom 14370 minregex2 43030 |
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