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Theorem carden 10545
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.)

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 10464 . . . . . 6 (𝐴 ∈ 𝐢 β†’ 𝐴 ∈ dom card)
21ad2antrr 723 . . . . 5 (((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) ∧ (cardβ€˜π΄) = (cardβ€˜π΅)) β†’ 𝐴 ∈ dom card)
3 cardid2 9947 . . . . 5 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
4 ensym 8998 . . . . 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 484 . . . . 5 (((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) ∧ (cardβ€˜π΄) = (cardβ€˜π΅)) β†’ (cardβ€˜π΄) = (cardβ€˜π΅))
7 numth3 10464 . . . . . . 7 (𝐡 ∈ 𝐷 β†’ 𝐡 ∈ dom card)
87ad2antlr 724 . . . . . 6 (((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) ∧ (cardβ€˜π΄) = (cardβ€˜π΅)) β†’ 𝐡 ∈ dom card)
98cardidd 10543 . . . . 5 (((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) ∧ (cardβ€˜π΄) = (cardβ€˜π΅)) β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
106, 9eqbrtrd 5163 . . . 4 (((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) ∧ (cardβ€˜π΄) = (cardβ€˜π΅)) β†’ (cardβ€˜π΄) β‰ˆ 𝐡)
11 entr 9001 . . . 4 ((𝐴 β‰ˆ (cardβ€˜π΄) ∧ (cardβ€˜π΄) β‰ˆ 𝐡) β†’ 𝐴 β‰ˆ 𝐡)
125, 10, 11syl2anc 583 . . 3 (((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) ∧ (cardβ€˜π΄) = (cardβ€˜π΅)) β†’ 𝐴 β‰ˆ 𝐡)
1312ex 412 . 2 ((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) β†’ ((cardβ€˜π΄) = (cardβ€˜π΅) β†’ 𝐴 β‰ˆ 𝐡))
14 carden2b 9961 . 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 395   = wceq 1533   ∈ wcel 2098   class class class wbr 5141  dom cdm 5669  β€˜cfv 6536   β‰ˆ 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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