Theorem ficard 10522

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 8956	. . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
2		carden 10508	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → ((card‘𝐴) = (card‘𝑥) ↔ 𝐴 ≈ 𝑥))
3		cardnn 9921	. . . . . . . 8 ⊢ (𝑥 ∈ ω → (card‘𝑥) = 𝑥)
4		eqtr 2782	. . . . . . . . 9 ⊢ (((card‘𝐴) = (card‘𝑥) ∧ (card‘𝑥) = 𝑥) → (card‘𝐴) = 𝑥)
5	4	expcom 417	. . . . . . . 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 2857	. . . . . . 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 485	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → ((card‘𝐴) = (card‘𝑥) → (card‘𝐴) ∈ ω))
10	2, 9	sylbird 262	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ ω) → (𝐴 ≈ 𝑥 → (card‘𝐴) ∈ ω))
11	10	rexlimdva 3163	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → (card‘𝐴) ∈ ω))
12	1, 11	biimtrid 244	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin → (card‘𝐴) ∈ ω))
13		cardnn 9921	. . . . . . . 8 ⊢ ((card‘𝐴) ∈ ω → (card‘(card‘𝐴)) = (card‘𝐴))
14	13	eqcomd 2768	. . . . . . 7 ⊢ ((card‘𝐴) ∈ ω → (card‘𝐴) = (card‘(card‘𝐴)))
15	14	adantl 485	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (card‘𝐴) ∈ ω) → (card‘𝐴) = (card‘(card‘𝐴)))
16		carden 10508	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (card‘𝐴) ∈ ω) → ((card‘𝐴) = (card‘(card‘𝐴)) ↔ 𝐴 ≈ (card‘𝐴)))
17	15, 16	mpbid 234	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (card‘𝐴) ∈ ω) → 𝐴 ≈ (card‘𝐴))
18	17	ex 416	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) ∈ ω → 𝐴 ≈ (card‘𝐴)))
19	18	ancld 558	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) ∈ ω → ((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴))))
20		breq2 5104	. . . . 5 ⊢ (𝑥 = (card‘𝐴) → (𝐴 ≈ 𝑥 ↔ 𝐴 ≈ (card‘𝐴)))
21	20	rspcev 3581	. . . 4 ⊢ (((card‘𝐴) ∈ ω ∧ 𝐴 ≈ (card‘𝐴)) → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
22	21, 1	sylibr 236	. . 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 214	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin ↔ (card‘𝐴) ∈ ω))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∃wrex 3086   class class class wbr 5100  ‘cfv 6521  ωcom 7846   ≈ cen 8924  Fincfn 8927  cardccrd 9893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-ac2 10420

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-1o 8437  df-er 8678  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-card 9897  df-ac 10072

This theorem is referenced by:  cfpwsdom  10542