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Mirrors > Home > MPE Home > Th. List > canth3 | Structured version Visualization version GIF version |
Description: Cantor's theorem in terms of cardinals. This theorem tells us that no matter how large a cardinal number is, there is a still larger cardinal number. Theorem 18.12 of [Monk1] p. 133. (Contributed by NM, 5-Nov-2003.) |
Ref | Expression |
---|---|
canth3 | β’ (π΄ β π β (cardβπ΄) β (cardβπ« π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | canth2g 9078 | . 2 β’ (π΄ β π β π΄ βΊ π« π΄) | |
2 | pwexg 5334 | . . 3 β’ (π΄ β π β π« π΄ β V) | |
3 | cardsdom 10496 | . . 3 β’ ((π΄ β π β§ π« π΄ β V) β ((cardβπ΄) β (cardβπ« π΄) β π΄ βΊ π« π΄)) | |
4 | 2, 3 | mpdan 686 | . 2 β’ (π΄ β π β ((cardβπ΄) β (cardβπ« π΄) β π΄ βΊ π« π΄)) |
5 | 1, 4 | mpbird 257 | 1 β’ (π΄ β π β (cardβπ΄) β (cardβπ« π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β wcel 2107 Vcvv 3444 π« cpw 4561 class class class wbr 5106 βcfv 6497 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-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-ac2 10404 |
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-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 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-iun 4957 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-se 5590 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-pred 6254 df-ord 6321 df-on 6322 df-suc 6324 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-isom 6506 df-riota 7314 df-ov 7361 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-card 9880 df-ac 10057 |
This theorem is referenced by: (None) |
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