Theorem rp-isfinite5 41022

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 RP, 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 6769	. . . 4 ⊢ (♯‘𝐴) ∈ V
2		hashcl 13999	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
3		isfinite4 14005	. . . . . 6 ⊢ (𝐴 ∈ Fin ↔ (1...(♯‘𝐴)) ≈ 𝐴)
4	3	biimpi 215	. . . . 5 ⊢ (𝐴 ∈ Fin → (1...(♯‘𝐴)) ≈ 𝐴)
5	2, 4	jca 511	. . . 4 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ0 ∧ (1...(♯‘𝐴)) ≈ 𝐴))
6		eleq1 2826	. . . . . 6 ⊢ (𝑛 = (♯‘𝐴) → (𝑛 ∈ ℕ0 ↔ (♯‘𝐴) ∈ ℕ0))
7		oveq2 7263	. . . . . . 7 ⊢ (𝑛 = (♯‘𝐴) → (1...𝑛) = (1...(♯‘𝐴)))
8	7	breq1d 5080	. . . . . 6 ⊢ (𝑛 = (♯‘𝐴) → ((1...𝑛) ≈ 𝐴 ↔ (1...(♯‘𝐴)) ≈ 𝐴))
9	6, 8	anbi12d 630	. . . . 5 ⊢ (𝑛 = (♯‘𝐴) → ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) ↔ ((♯‘𝐴) ∈ ℕ0 ∧ (1...(♯‘𝐴)) ≈ 𝐴)))
10	9	spcegv 3526	. . . 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 3069	. . 3 ⊢ (∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴 ↔ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴))
13	11, 12	sylibr 233	. 2 ⊢ (𝐴 ∈ Fin → ∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴)
14		hasheni 13990	. . . . . . 7 ⊢ ((1...𝑛) ≈ 𝐴 → (♯‘(1...𝑛)) = (♯‘𝐴))
15	14	eqcomd 2744	. . . . . 6 ⊢ ((1...𝑛) ≈ 𝐴 → (♯‘𝐴) = (♯‘(1...𝑛)))
16		hashfz1 13988	. . . . . 6 ⊢ (𝑛 ∈ ℕ0 → (♯‘(1...𝑛)) = 𝑛)
17		ovex 7288	. . . . . . 7 ⊢ (1...(♯‘𝐴)) ∈ V
18		eqtr 2761	. . . . . . 7 ⊢ (((♯‘𝐴) = (♯‘(1...𝑛)) ∧ (♯‘(1...𝑛)) = 𝑛) → (♯‘𝐴) = 𝑛)
19		oveq2 7263	. . . . . . . 8 ⊢ ((♯‘𝐴) = 𝑛 → (1...(♯‘𝐴)) = (1...𝑛))
20		eqeng 8729	. . . . . . . 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 596	. . . . 5 ⊢ ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) → (1...(♯‘𝐴)) ≈ (1...𝑛))
24		entr 8747	. . . . 5 ⊢ (((1...(♯‘𝐴)) ≈ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴) → (1...(♯‘𝐴)) ≈ 𝐴)
25	23, 24	sylancom 587	. . . 4 ⊢ ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) → (1...(♯‘𝐴)) ≈ 𝐴)
26	25, 3	sylibr 233	. . 3 ⊢ ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) → 𝐴 ∈ Fin)
27	26	rexlimiva 3209	. 2 ⊢ (∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴 → 𝐴 ∈ Fin)
28	13, 27	impbii 208	1 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴)

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∃wrex 3064  Vcvv 3422   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255   ≈ cen 8688  Fincfn 8691  1c1 10803  ℕ₀cn0 12163  ...cfz 13168  ♯chash 13972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-hash 13973

This theorem is referenced by:  rp-isfinite6  41023