Theorem ficard 10602

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 9014	. . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
2		carden 10588	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → ((card‘𝐴) = (card‘𝑥) ↔ 𝐴 ≈ 𝑥))
3		cardnn 10000	. . . . . . . 8 ⊢ (𝑥 ∈ ω → (card‘𝑥) = 𝑥)
4		eqtr 2757	. . . . . . . . 9 ⊢ (((card‘𝐴) = (card‘𝑥) ∧ (card‘𝑥) = 𝑥) → (card‘𝐴) = 𝑥)
5	4	expcom 413	. . . . . . . 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 2833	. . . . . . 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 481	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → ((card‘𝐴) = (card‘𝑥) → (card‘𝐴) ∈ ω))
10	2, 9	sylbird 260	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → (𝐴 ≈ 𝑥 → (card‘𝐴) ∈ ω))
11	10	rexlimdva 3152	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → (card‘𝐴) ∈ ω))
12	1, 11	biimtrid 242	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin → (card‘𝐴) ∈ ω))
13		cardnn 10000	. . . . . . . 8 ⊢ ((card‘𝐴) ∈ ω → (card‘(card‘𝐴)) = (card‘𝐴))
14	13	eqcomd 2740	. . . . . . 7 ⊢ ((card‘𝐴) ∈ ω → (card‘𝐴) = (card‘(card‘𝐴)))
15	14	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (card‘𝐴) ∈ ω) → (card‘𝐴) = (card‘(card‘𝐴)))
16		carden 10588	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (card‘𝐴) ∈ ω) → ((card‘𝐴) = (card‘(card‘𝐴)) ↔ 𝐴 ≈ (card‘𝐴)))
17	15, 16	mpbid 232	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (card‘𝐴) ∈ ω) → 𝐴 ≈ (card‘𝐴))
18	17	ex 412	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) ∈ ω → 𝐴 ≈ (card‘𝐴)))
19	18	ancld 550	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) ∈ ω → ((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴))))
20		breq2 5151	. . . . 5 ⊢ (𝑥 = (card‘𝐴) → (𝐴 ≈ 𝑥 ↔ 𝐴 ≈ (card‘𝐴)))
21	20	rspcev 3621	. . . 4 ⊢ (((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴)) → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
22	21, 1	sylibr 234	. . 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 212	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin ↔ (card‘𝐴) ∈ ω))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ∃wrex 3067   class class class wbr 5147  ‘cfv 6562  ωcom 7886   ≈ cen 8980  Fincfn 8983  cardccrd 9972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-ac2 10500

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-1o 8504  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-card 9976  df-ac 10153

This theorem is referenced by:  cfpwsdom  10621