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Mirrors > Home > MPE Home > Th. List > cardeq0 | Structured version Visualization version GIF version |
Description: Only the empty set has cardinality zero. (Contributed by NM, 23-Apr-2004.) |
Ref | Expression |
---|---|
cardeq0 | β’ (π΄ β π β ((cardβπ΄) = β β π΄ = β )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5309 | . . 3 β’ β β V | |
2 | carden 10580 | . . 3 β’ ((π΄ β π β§ β β V) β ((cardβπ΄) = (cardββ ) β π΄ β β )) | |
3 | 1, 2 | mpan2 689 | . 2 β’ (π΄ β π β ((cardβπ΄) = (cardββ ) β π΄ β β )) |
4 | card0 9987 | . . 3 β’ (cardββ ) = β | |
5 | 4 | eqeq2i 2740 | . 2 β’ ((cardβπ΄) = (cardββ ) β (cardβπ΄) = β ) |
6 | en0 9042 | . 2 β’ (π΄ β β β π΄ = β ) | |
7 | 3, 5, 6 | 3bitr3g 312 | 1 β’ (π΄ β π β ((cardβπ΄) = β β π΄ = β )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1533 β wcel 2098 Vcvv 3471 β c0 4324 class class class wbr 5150 βcfv 6551 β cen 8965 cardccrd 9964 |
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 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-ac2 10492 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-er 8729 df-en 8969 df-card 9968 df-ac 10145 |
This theorem is referenced by: tskcard 10810 |
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