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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 14311 and the finite-set-only hashen 14310.
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 3771). 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 9889). (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 10464 | . . . . . 6 β’ (π΄ β πΆ β π΄ β dom card) | |
2 | 1 | ad2antrr 723 | . . . . 5 β’ (((π΄ β πΆ β§ π΅ β π·) β§ (cardβπ΄) = (cardβπ΅)) β π΄ β dom card) |
3 | cardid2 9947 | . . . . 5 β’ (π΄ β dom card β (cardβπ΄) β π΄) | |
4 | ensym 8998 | . . . . 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 484 | . . . . 5 β’ (((π΄ β πΆ β§ π΅ β π·) β§ (cardβπ΄) = (cardβπ΅)) β (cardβπ΄) = (cardβπ΅)) | |
7 | numth3 10464 | . . . . . . 7 β’ (π΅ β π· β π΅ β dom card) | |
8 | 7 | ad2antlr 724 | . . . . . 6 β’ (((π΄ β πΆ β§ π΅ β π·) β§ (cardβπ΄) = (cardβπ΅)) β π΅ β dom card) |
9 | 8 | cardidd 10543 | . . . . 5 β’ (((π΄ β πΆ β§ π΅ β π·) β§ (cardβπ΄) = (cardβπ΅)) β (cardβπ΅) β π΅) |
10 | 6, 9 | eqbrtrd 5163 | . . . 4 β’ (((π΄ β πΆ β§ π΅ β π·) β§ (cardβπ΄) = (cardβπ΅)) β (cardβπ΄) β π΅) |
11 | entr 9001 | . . . 4 β’ ((π΄ β (cardβπ΄) β§ (cardβπ΄) β π΅) β π΄ β π΅) | |
12 | 5, 10, 11 | syl2anc 583 | . . 3 β’ (((π΄ β πΆ β§ π΅ β π·) β§ (cardβπ΄) = (cardβπ΅)) β π΄ β π΅) |
13 | 12 | ex 412 | . 2 β’ ((π΄ β πΆ β§ π΅ β π·) β ((cardβπ΄) = (cardβπ΅) β π΄ β π΅)) |
14 | carden2b 9961 | . 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 395 = wceq 1533 β wcel 2098 class class class wbr 5141 dom cdm 5669 βcfv 6536 β cen 8935 cardccrd 9929 |
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-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-ac2 10457 |
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-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 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-iun 4992 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-se 5625 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-pred 6293 df-ord 6360 df-on 6361 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-er 8702 df-en 8939 df-card 9933 df-ac 10110 |
This theorem is referenced by: cardeq0 10546 ficard 10559 |
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