Theorem ficard 10322

Description: A set is finite iff its cardinal is a natural number. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
ficard	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin ↔ (card‘𝐴) ∈ ω))

Proof of Theorem ficard

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isfi 8747	. . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
2		carden 10308	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → ((card‘𝐴) = (card‘𝑥) ↔ 𝐴 ≈ 𝑥))
3		cardnn 9722	. . . . . . . 8 ⊢ (𝑥 ∈ ω → (card‘𝑥) = 𝑥)
4		eqtr 2763	. . . . . . . . 9 ⊢ (((card‘𝐴) = (card‘𝑥) ∧ (card‘𝑥) = 𝑥) → (card‘𝐴) = 𝑥)
5	4	expcom 414	. . . . . . . 8 ⊢ ((card‘𝑥) = 𝑥 → ((card‘𝐴) = (card‘𝑥) → (card‘𝐴) = 𝑥))
6	3, 5	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ ω → ((card‘𝐴) = (card‘𝑥) → (card‘𝐴) = 𝑥))
7		eleq1a 2836	. . . . . . 7 ⊢ (𝑥 ∈ ω → ((card‘𝐴) = 𝑥 → (card‘𝐴) ∈ ω))
8	6, 7	syld 47	. . . . . 6 ⊢ (𝑥 ∈ ω → ((card‘𝐴) = (card‘𝑥) → (card‘𝐴) ∈ ω))
9	8	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → ((card‘𝐴) = (card‘𝑥) → (card‘𝐴) ∈ ω))
10	2, 9	sylbird 259	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → (𝐴 ≈ 𝑥 → (card‘𝐴) ∈ ω))
11	10	rexlimdva 3215	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → (card‘𝐴) ∈ ω))
12	1, 11	syl5bi 241	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin → (card‘𝐴) ∈ ω))
13		cardnn 9722	. . . . . . . 8 ⊢ ((card‘𝐴) ∈ ω → (card‘(card‘𝐴)) = (card‘𝐴))
14	13	eqcomd 2746	. . . . . . 7 ⊢ ((card‘𝐴) ∈ ω → (card‘𝐴) = (card‘(card‘𝐴)))
15	14	adantl 482	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (card‘𝐴) ∈ ω) → (card‘𝐴) = (card‘(card‘𝐴)))
16		carden 10308	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (card‘𝐴) ∈ ω) → ((card‘𝐴) = (card‘(card‘𝐴)) ↔ 𝐴 ≈ (card‘𝐴)))
17	15, 16	mpbid 231	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (card‘𝐴) ∈ ω) → 𝐴 ≈ (card‘𝐴))
18	17	ex 413	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) ∈ ω → 𝐴 ≈ (card‘𝐴)))
19	18	ancld 551	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) ∈ ω → ((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴))))
20		breq2 5083	. . . . 5 ⊢ (𝑥 = (card‘𝐴) → (𝐴 ≈ 𝑥 ↔ 𝐴 ≈ (card‘𝐴)))
21	20	rspcev 3561	. . . 4 ⊢ (((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴)) → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
22	21, 1	sylibr 233	. . 3 ⊢ (((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴)) → 𝐴 ∈ Fin)
23	19, 22	syl6 35	. 2 ⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) ∈ ω → 𝐴 ∈ Fin))
24	12, 23	impbid 211	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin ↔ (card‘𝐴) ∈ ω))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1542   ∈ wcel 2110  ∃wrex 3067   class class class wbr 5079  ‘cfv 6432  ωcom 7706   ≈ cen 8713  Fincfn 8716  cardccrd 9694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-ac2 10220

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-se 5546  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-isom 6441  df-riota 7228  df-ov 7274  df-om 7707  df-2nd 7825  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-1o 8288  df-er 8481  df-en 8717  df-dom 8718  df-sdom 8719  df-fin 8720  df-card 9698  df-ac 9873

This theorem is referenced by:  cfpwsdom  10341