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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 10551, 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 9967 | . 2 β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΄) β (cardβπ΅) β π΄ βΌ π΅)) | |
2 | brdom2 8980 | . . 3 β’ (π΄ βΌ π΅ β (π΄ βΊ π΅ β¨ π΄ β π΅)) | |
3 | cardon 9941 | . . . . . . . 8 β’ (cardβπ΄) β On | |
4 | 3 | onelssi 6473 | . . . . . . 7 β’ ((cardβπ΅) β (cardβπ΄) β (cardβπ΅) β (cardβπ΄)) |
5 | carddomi2 9967 | . . . . . . . 8 β’ ((π΅ β dom card β§ π΄ β dom card) β ((cardβπ΅) β (cardβπ΄) β π΅ βΌ π΄)) | |
6 | 5 | ancoms 458 | . . . . . . 7 β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΅) β (cardβπ΄) β π΅ βΌ π΄)) |
7 | domnsym 9101 | . . . . . . 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 9941 | . . . . . 6 β’ (cardβπ΅) β On | |
11 | ontri1 6392 | . . . . . 6 β’ (((cardβπ΄) β On β§ (cardβπ΅) β On) β ((cardβπ΄) β (cardβπ΅) β Β¬ (cardβπ΅) β (cardβπ΄))) | |
12 | 3, 10, 11 | mp2an 689 | . . . . 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 9964 | . . . . . 6 β’ (π΄ β π΅ β (cardβπ΄) = (cardβπ΅)) | |
15 | eqimss 4035 | . . . . . 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 856 | . . 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 395 β¨ wo 844 = wceq 1533 β wcel 2098 β wss 3943 class class class wbr 5141 dom cdm 5669 Oncon0 6358 βcfv 6537 β cen 8938 βΌ cdom 8939 βΊ csdm 8940 cardccrd 9932 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-card 9936 |
This theorem is referenced by: carduni 9978 carden2 9984 cardsdom2 9985 domtri2 9986 infxpidm2 10014 cardaleph 10086 infenaleph 10088 alephinit 10092 ficardun2 10199 ficardun2OLD 10200 ackbij2 10240 cfflb 10256 fin1a2lem9 10405 carddom 10551 pwfseqlem5 10660 hashdom 14344 minregex2 42862 |
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