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Theorem carden 10492
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.)

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 10411 . . . . . 6 (𝐴 ∈ 𝐢 β†’ 𝐴 ∈ dom card)
21ad2antrr 725 . . . . 5 (((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) ∧ (cardβ€˜π΄) = (cardβ€˜π΅)) β†’ 𝐴 ∈ dom card)
3 cardid2 9894 . . . . 5 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
4 ensym 8946 . . . . 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 486 . . . . 5 (((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) ∧ (cardβ€˜π΄) = (cardβ€˜π΅)) β†’ (cardβ€˜π΄) = (cardβ€˜π΅))
7 numth3 10411 . . . . . . 7 (𝐡 ∈ 𝐷 β†’ 𝐡 ∈ dom card)
87ad2antlr 726 . . . . . 6 (((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) ∧ (cardβ€˜π΄) = (cardβ€˜π΅)) β†’ 𝐡 ∈ dom card)
98cardidd 10490 . . . . 5 (((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) ∧ (cardβ€˜π΄) = (cardβ€˜π΅)) β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
106, 9eqbrtrd 5128 . . . 4 (((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) ∧ (cardβ€˜π΄) = (cardβ€˜π΅)) β†’ (cardβ€˜π΄) β‰ˆ 𝐡)
11 entr 8949 . . . 4 ((𝐴 β‰ˆ (cardβ€˜π΄) ∧ (cardβ€˜π΄) β‰ˆ 𝐡) β†’ 𝐴 β‰ˆ 𝐡)
125, 10, 11syl2anc 585 . . 3 (((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) ∧ (cardβ€˜π΄) = (cardβ€˜π΅)) β†’ 𝐴 β‰ˆ 𝐡)
1312ex 414 . 2 ((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) β†’ ((cardβ€˜π΄) = (cardβ€˜π΅) β†’ 𝐴 β‰ˆ 𝐡))
14 carden2b 9908 . 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 397   = wceq 1542   ∈ wcel 2107   class class class wbr 5106  dom cdm 5634  β€˜cfv 6497   β‰ˆ 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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