Theorem fidcenum 6985

Description: A set is finite if and only if it has decidable equality and is finitely enumerable. Proposition 8.1.11 of [AczelRathjen], p. 72. The definition of "finitely enumerable" as ∃𝑛 ∈ ω∃𝑓𝑓:𝑛–onto→𝐴 is Definition 8.1.4 of [AczelRathjen], p. 71. (Contributed by Jim Kingdon, 19-Oct-2022.)

Assertion

Ref	Expression
fidcenum	⊢ (𝐴 ∈ Fin ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛–onto→𝐴))

Distinct variable group:   𝐴,𝑓,𝑛,𝑥,𝑦

Proof of Theorem fidcenum

Step	Hyp	Ref	Expression
1		fidcenumlemim 6981	. 2 ⊢ (𝐴 ∈ Fin → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛–onto→𝐴))
2		simpll 527	. . . . . . 7 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑛 ∈ ω) ∧ 𝑓:𝑛–onto→𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
3		simpr 110	. . . . . . 7 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑛 ∈ ω) ∧ 𝑓:𝑛–onto→𝐴) → 𝑓:𝑛–onto→𝐴)
4		simplr 528	. . . . . . 7 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑛 ∈ ω) ∧ 𝑓:𝑛–onto→𝐴) → 𝑛 ∈ ω)
5	2, 3, 4	fidcenumlemr 6984	. . . . . 6 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑛 ∈ ω) ∧ 𝑓:𝑛–onto→𝐴) → 𝐴 ∈ Fin)
6	5	ex 115	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑛 ∈ ω) → (𝑓:𝑛–onto→𝐴 → 𝐴 ∈ Fin))
7	6	exlimdv 1830	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝑛 ∈ ω) → (∃𝑓 𝑓:𝑛–onto→𝐴 → 𝐴 ∈ Fin))
8	7	rexlimdva 2607	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 → (∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛–onto→𝐴 → 𝐴 ∈ Fin))
9	8	imp 124	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛–onto→𝐴) → 𝐴 ∈ Fin)
10	1, 9	impbii 126	1 ⊢ (𝐴 ∈ Fin ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛–onto→𝐴))

Syntax hints:   ∧ wa 104   ↔ wb 105  DECID wdc 835  ∃wex 1503   ∈ wcel 2160  ∀wral 2468  ∃wrex 2469  ωcom 4607  –onto→wfo 5233  Fincfn 6766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1o 6441  df-er 6559  df-en 6767  df-fin 6769

This theorem is referenced by:  finct  7145  ctinf  12481