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Mirrors > Home > MPE Home > Th. List > carden | Structured version Visualization version GIF version |
Description: Two sets are equinumerous
iff their cardinal numbers are equal. This
important theorem expresses the essential concept behind
"cardinality" or
"size". This theorem appears as Proposition 10.10 of [TakeutiZaring]
p. 85, Theorem 7P of [Enderton] p. 197,
and Theorem 9 of [Suppes] p. 242
(among others). The Axiom of Choice is required for its proof. Related
theorems are hasheni 14254 and the finite-set-only hashen 14253.
This theorem is also known as Hume's Principle. Gottlob Frege's two-volume Grundgesetze der Arithmetik used his Basic Law V to prove this theorem. Unfortunately Basic Law V caused Frege's system to be inconsistent because it was subject to Russell's paradox (see ru 3739). Later scholars have found that Frege primarily used Basic Law V to Hume's Principle. If Basic Law V is replaced by Hume's Principle in Frege's system, much of Frege's work is restored. Grundgesetze der Arithmetik, once Basic Law V is replaced, proves "Frege's theorem" (the Peano axioms of arithmetic can be derived in second-order logic from Hume's principle). See https://plato.stanford.edu/entries/frege-theorem . We take a different approach, using first-order logic and ZFC, to prove the Peano axioms of arithmetic. The theory of cardinality can also be developed without AC by introducing "card" as a primitive notion and stating this theorem as an axiom, as is done with the axiom for cardinal numbers in [Suppes] p. 111. Finally, if we allow the Axiom of Regularity, we can avoid AC by defining the cardinal number of a set as the set of all sets equinumerous to it and having the least possible rank (see karden 9836). (Contributed by NM, 22-Oct-2003.) |
Ref | Expression |
---|---|
carden | β’ ((π΄ β πΆ β§ π΅ β π·) β ((cardβπ΄) = (cardβπ΅) β π΄ β π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numth3 10411 | . . . . . 6 β’ (π΄ β πΆ β π΄ β dom card) | |
2 | 1 | ad2antrr 725 | . . . . 5 β’ (((π΄ β πΆ β§ π΅ β π·) β§ (cardβπ΄) = (cardβπ΅)) β π΄ β dom card) |
3 | cardid2 9894 | . . . . 5 β’ (π΄ β dom card β (cardβπ΄) β π΄) | |
4 | ensym 8946 | . . . . 5 β’ ((cardβπ΄) β π΄ β π΄ β (cardβπ΄)) | |
5 | 2, 3, 4 | 3syl 18 | . . . 4 β’ (((π΄ β πΆ β§ π΅ β π·) β§ (cardβπ΄) = (cardβπ΅)) β π΄ β (cardβπ΄)) |
6 | simpr 486 | . . . . 5 β’ (((π΄ β πΆ β§ π΅ β π·) β§ (cardβπ΄) = (cardβπ΅)) β (cardβπ΄) = (cardβπ΅)) | |
7 | numth3 10411 | . . . . . . 7 β’ (π΅ β π· β π΅ β dom card) | |
8 | 7 | ad2antlr 726 | . . . . . 6 β’ (((π΄ β πΆ β§ π΅ β π·) β§ (cardβπ΄) = (cardβπ΅)) β π΅ β dom card) |
9 | 8 | cardidd 10490 | . . . . 5 β’ (((π΄ β πΆ β§ π΅ β π·) β§ (cardβπ΄) = (cardβπ΅)) β (cardβπ΅) β π΅) |
10 | 6, 9 | eqbrtrd 5128 | . . . 4 β’ (((π΄ β πΆ β§ π΅ β π·) β§ (cardβπ΄) = (cardβπ΅)) β (cardβπ΄) β π΅) |
11 | entr 8949 | . . . 4 β’ ((π΄ β (cardβπ΄) β§ (cardβπ΄) β π΅) β π΄ β π΅) | |
12 | 5, 10, 11 | syl2anc 585 | . . 3 β’ (((π΄ β πΆ β§ π΅ β π·) β§ (cardβπ΄) = (cardβπ΅)) β π΄ β π΅) |
13 | 12 | ex 414 | . 2 β’ ((π΄ β πΆ β§ π΅ β π·) β ((cardβπ΄) = (cardβπ΅) β π΄ β π΅)) |
14 | carden2b 9908 | . 2 β’ (π΄ β π΅ β (cardβπ΄) = (cardβπ΅)) | |
15 | 13, 14 | impbid1 224 | 1 β’ ((π΄ β πΆ β§ π΅ β π·) β ((cardβπ΄) = (cardβπ΅) β π΄ β π΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 class class class wbr 5106 dom cdm 5634 βcfv 6497 β cen 8883 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-card 9880 df-ac 10057 |
This theorem is referenced by: cardeq0 10493 ficard 10506 |
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