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Theorem carden 10580
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 14345 and the finite-set-only hashen 14344.

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 3775). 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 9924). (Contributed by NM, 22-Oct-2003.)

Assertion
Ref Expression
carden ((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) β†’ ((cardβ€˜π΄) = (cardβ€˜π΅) ↔ 𝐴 β‰ˆ 𝐡))

Proof of Theorem carden
StepHypRef Expression
1 numth3 10499 . . . . . 6 (𝐴 ∈ 𝐢 β†’ 𝐴 ∈ dom card)
21ad2antrr 724 . . . . 5 (((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) ∧ (cardβ€˜π΄) = (cardβ€˜π΅)) β†’ 𝐴 ∈ dom card)
3 cardid2 9982 . . . . 5 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
4 ensym 9028 . . . . 5 ((cardβ€˜π΄) β‰ˆ 𝐴 β†’ 𝐴 β‰ˆ (cardβ€˜π΄))
52, 3, 43syl 18 . . . 4 (((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) ∧ (cardβ€˜π΄) = (cardβ€˜π΅)) β†’ 𝐴 β‰ˆ (cardβ€˜π΄))
6 simpr 483 . . . . 5 (((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) ∧ (cardβ€˜π΄) = (cardβ€˜π΅)) β†’ (cardβ€˜π΄) = (cardβ€˜π΅))
7 numth3 10499 . . . . . . 7 (𝐡 ∈ 𝐷 β†’ 𝐡 ∈ dom card)
87ad2antlr 725 . . . . . 6 (((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) ∧ (cardβ€˜π΄) = (cardβ€˜π΅)) β†’ 𝐡 ∈ dom card)
98cardidd 10578 . . . . 5 (((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) ∧ (cardβ€˜π΄) = (cardβ€˜π΅)) β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
106, 9eqbrtrd 5172 . . . 4 (((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) ∧ (cardβ€˜π΄) = (cardβ€˜π΅)) β†’ (cardβ€˜π΄) β‰ˆ 𝐡)
11 entr 9031 . . . 4 ((𝐴 β‰ˆ (cardβ€˜π΄) ∧ (cardβ€˜π΄) β‰ˆ 𝐡) β†’ 𝐴 β‰ˆ 𝐡)
125, 10, 11syl2anc 582 . . 3 (((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) ∧ (cardβ€˜π΄) = (cardβ€˜π΅)) β†’ 𝐴 β‰ˆ 𝐡)
1312ex 411 . 2 ((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) β†’ ((cardβ€˜π΄) = (cardβ€˜π΅) β†’ 𝐴 β‰ˆ 𝐡))
14 carden2b 9996 . 2 (𝐴 β‰ˆ 𝐡 β†’ (cardβ€˜π΄) = (cardβ€˜π΅))
1513, 14impbid1 224 1 ((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) β†’ ((cardβ€˜π΄) = (cardβ€˜π΅) ↔ 𝐴 β‰ˆ 𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   class class class wbr 5150  dom cdm 5680  β€˜cfv 6551   β‰ˆ 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:  cardeq0  10581  ficard  10594
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