Theorem fin0or 7118

Description: A finite set is either empty or inhabited. (Contributed by Jim Kingdon, 30-Sep-2021.)

Assertion

Ref	Expression
fin0or	⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ∃𝑥 𝑥 ∈ 𝐴))

Distinct variable group:   𝑥,𝐴

Proof of Theorem fin0or

Dummy variables 𝑓 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isfi 6977	. . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
2	1	biimpi 120	. 2 ⊢ (𝐴 ∈ Fin → ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
3		nn0suc 4708	. . . 4 ⊢ (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
4	3	ad2antrl 490	. . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → (𝑛 = ∅ ∨ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
5		simplrr 538	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑛 = ∅) → 𝐴 ≈ 𝑛)
6		simpr 110	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑛 = ∅) → 𝑛 = ∅)
7	5, 6	breqtrd 4119	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑛 = ∅) → 𝐴 ≈ ∅)
8		en0 7012	. . . . . 6 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
9	7, 8	sylib 122	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑛 = ∅) → 𝐴 = ∅)
10	9	ex 115	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → (𝑛 = ∅ → 𝐴 = ∅))
11		simplrr 538	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) → 𝐴 ≈ 𝑛)
12	11	adantr 276	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐴 ≈ 𝑛)
13	12	ensymd 7000	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝑛 ≈ 𝐴)
14		bren 6960	. . . . . . . 8 ⊢ (𝑛 ≈ 𝐴 ↔ ∃𝑓 𝑓:𝑛–1-1-onto→𝐴)
15	13, 14	sylib 122	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → ∃𝑓 𝑓:𝑛–1-1-onto→𝐴)
16		f1of 5592	. . . . . . . . . 10 ⊢ (𝑓:𝑛–1-1-onto→𝐴 → 𝑓:𝑛⟶𝐴)
17	16	adantl 277	. . . . . . . . 9 ⊢ (((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) ∧ 𝑓:𝑛–1-1-onto→𝐴) → 𝑓:𝑛⟶𝐴)
18		sucidg 4519	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ω → 𝑚 ∈ suc 𝑚)
19	18	ad3antlr 493	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) ∧ 𝑓:𝑛–1-1-onto→𝐴) → 𝑚 ∈ suc 𝑚)
20		simplr 529	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) ∧ 𝑓:𝑛–1-1-onto→𝐴) → 𝑛 = suc 𝑚)
21	19, 20	eleqtrrd 2311	. . . . . . . . 9 ⊢ (((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) ∧ 𝑓:𝑛–1-1-onto→𝐴) → 𝑚 ∈ 𝑛)
22	17, 21	ffvelcdmd 5791	. . . . . . . 8 ⊢ (((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) ∧ 𝑓:𝑛–1-1-onto→𝐴) → (𝑓‘𝑚) ∈ 𝐴)
23		elex2 2820	. . . . . . . 8 ⊢ ((𝑓‘𝑚) ∈ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴)
24	22, 23	syl 14	. . . . . . 7 ⊢ (((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) ∧ 𝑓:𝑛–1-1-onto→𝐴) → ∃𝑥 𝑥 ∈ 𝐴)
25	15, 24	exlimddv 1947	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → ∃𝑥 𝑥 ∈ 𝐴)
26	25	ex 115	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) → (𝑛 = suc 𝑚 → ∃𝑥 𝑥 ∈ 𝐴))
27	26	rexlimdva 2651	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → (∃𝑚 ∈ ω 𝑛 = suc 𝑚 → ∃𝑥 𝑥 ∈ 𝐴))
28	10, 27	orim12d 794	. . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → ((𝑛 = ∅ ∨ ∃𝑚 ∈ ω 𝑛 = suc 𝑚) → (𝐴 = ∅ ∨ ∃𝑥 𝑥 ∈ 𝐴)))
29	4, 28	mpd 13	. 2 ⊢ ((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → (𝐴 = ∅ ∨ ∃𝑥 𝑥 ∈ 𝐴))
30	2, 29	rexlimddv 2656	1 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ∃𝑥 𝑥 ∈ 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 716   = wceq 1398  ∃wex 1541   ∈ wcel 2202  ∃wrex 2512  ∅c0 3496   class class class wbr 4093  suc csuc 4468  ωcom 4694  ⟶wf 5329  –1-1-onto→wf1o 5332  ‘cfv 5333   ≈ cen 6950  Fincfn 6952

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-id 4396  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-er 6745  df-en 6953  df-fin 6955

This theorem is referenced by:  xpfi  7167  fival  7212  fiubm  11138  lswex  11214  fsumcllem  12023  fprodcllem  12230  gsumwsubmcl  13642  gsumwmhm  13644