Theorem rp-isfinite5 38383

Description: A set is said to be finite if it can be put in one-to-one correspondence with all the natural numbers between 1 and some 𝑛 ∈ ℕ₀. (Contributed by Richard Penner, 3-Mar-2020.)

Assertion

Ref	Expression
rp-isfinite5	⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴)

Distinct variable group:   𝐴,𝑛

Proof of Theorem rp-isfinite5

Step	Hyp	Ref	Expression
1		fvex 6363	. . . 4 ⊢ (♯‘𝐴) ∈ V
2		hashcl 13359	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
3		isfinite4 13365	. . . . . 6 ⊢ (𝐴 ∈ Fin ↔ (1...(♯‘𝐴)) ≈ 𝐴)
4	3	biimpi 206	. . . . 5 ⊢ (𝐴 ∈ Fin → (1...(♯‘𝐴)) ≈ 𝐴)
5	2, 4	jca 555	. . . 4 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ0 ∧ (1...(♯‘𝐴)) ≈ 𝐴))
6		eleq1 2827	. . . . . 6 ⊢ (𝑛 = (♯‘𝐴) → (𝑛 ∈ ℕ0 ↔ (♯‘𝐴) ∈ ℕ0))
7		oveq2 6822	. . . . . . 7 ⊢ (𝑛 = (♯‘𝐴) → (1...𝑛) = (1...(♯‘𝐴)))
8	7	breq1d 4814	. . . . . 6 ⊢ (𝑛 = (♯‘𝐴) → ((1...𝑛) ≈ 𝐴 ↔ (1...(♯‘𝐴)) ≈ 𝐴))
9	6, 8	anbi12d 749	. . . . 5 ⊢ (𝑛 = (♯‘𝐴) → ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) ↔ ((♯‘𝐴) ∈ ℕ0 ∧ (1...(♯‘𝐴)) ≈ 𝐴)))
10	9	spcegv 3434	. . . 4 ⊢ ((♯‘𝐴) ∈ V → (((♯‘𝐴) ∈ ℕ0 ∧ (1...(♯‘𝐴)) ≈ 𝐴) → ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)))
11	1, 5, 10	mpsyl 68	. . 3 ⊢ (𝐴 ∈ Fin → ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴))
12		df-rex 3056	. . 3 ⊢ (∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴 ↔ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴))
13	11, 12	sylibr 224	. 2 ⊢ (𝐴 ∈ Fin → ∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴)
14		hasheni 13350	. . . . . . 7 ⊢ ((1...𝑛) ≈ 𝐴 → (♯‘(1...𝑛)) = (♯‘𝐴))
15	14	eqcomd 2766	. . . . . 6 ⊢ ((1...𝑛) ≈ 𝐴 → (♯‘𝐴) = (♯‘(1...𝑛)))
16		hashfz1 13348	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 → (♯‘(1...𝑛)) = 𝑛)
17		ovex 6842	. . . . . . 7 ⊢ (1...(♯‘𝐴)) ∈ V
18		eqtr 2779	. . . . . . 7 ⊢ (((♯‘𝐴) = (♯‘(1...𝑛)) ∧ (♯‘(1...𝑛)) = 𝑛) → (♯‘𝐴) = 𝑛)
19		oveq2 6822	. . . . . . . 8 ⊢ ((♯‘𝐴) = 𝑛 → (1...(♯‘𝐴)) = (1...𝑛))
20		eqeng 8157	. . . . . . . 8 ⊢ ((1...(♯‘𝐴)) ∈ V → ((1...(♯‘𝐴)) = (1...𝑛) → (1...(♯‘𝐴)) ≈ (1...𝑛)))
21	19, 20	syl5 34	. . . . . . 7 ⊢ ((1...(♯‘𝐴)) ∈ V → ((♯‘𝐴) = 𝑛 → (1...(♯‘𝐴)) ≈ (1...𝑛)))
22	17, 18, 21	mpsyl 68	. . . . . 6 ⊢ (((♯‘𝐴) = (♯‘(1...𝑛)) ∧ (♯‘(1...𝑛)) = 𝑛) → (1...(♯‘𝐴)) ≈ (1...𝑛))
23	15, 16, 22	syl2anr 496	. . . . 5 ⊢ ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) → (1...(♯‘𝐴)) ≈ (1...𝑛))
24		entr 8175	. . . . 5 ⊢ (((1...(♯‘𝐴)) ≈ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴) → (1...(♯‘𝐴)) ≈ 𝐴)
25	23, 24	sylancom 704	. . . 4 ⊢ ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) → (1...(♯‘𝐴)) ≈ 𝐴)
26	25, 3	sylibr 224	. . 3 ⊢ ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) → 𝐴 ∈ Fin)
27	26	rexlimiva 3166	. 2 ⊢ (∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴 → 𝐴 ∈ Fin)
28	13, 27	impbii 199	1 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴)

Syntax hints:   ↔ wb 196   ∧ wa 383   = wceq 1632  ∃wex 1853   ∈ wcel 2139  ∃wrex 3051  Vcvv 3340   class class class wbr 4804  ‘cfv 6049  (class class class)co 6814   ≈ cen 8120  Fincfn 8123  1c1 10149  ℕ₀cn0 11504  ...cfz 12539  ♯chash 13331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-card 8975  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-n0 11505  df-z 11590  df-uz 11900  df-fz 12540  df-hash 13332

This theorem is referenced by:  rp-isfinite6  38384