Theorem fin0or 6535

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 6411	. . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
2	1	biimpi 118	. 2 ⊢ (𝐴 ∈ Fin → ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
3		nn0suc 4385	. . . 4 ⊢ (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
4	3	ad2antrl 474	. . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → (𝑛 = ∅ ∨ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
5		simplrr 503	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑛 = ∅) → 𝐴 ≈ 𝑛)
6		simpr 108	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑛 = ∅) → 𝑛 = ∅)
7	5, 6	breqtrd 3838	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑛 = ∅) → 𝐴 ≈ ∅)
8		en0 6445	. . . . . 6 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
9	7, 8	sylib 120	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑛 = ∅) → 𝐴 = ∅)
10	9	ex 113	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → (𝑛 = ∅ → 𝐴 = ∅))
11		simplrr 503	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) → 𝐴 ≈ 𝑛)
12	11	adantr 270	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐴 ≈ 𝑛)
13	12	ensymd 6433	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝑛 ≈ 𝐴)
14		bren 6397	. . . . . . . 8 ⊢ (𝑛 ≈ 𝐴 ↔ ∃𝑓 𝑓:𝑛–1-1-onto→𝐴)
15	13, 14	sylib 120	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → ∃𝑓 𝑓:𝑛–1-1-onto→𝐴)
16		f1of 5204	. . . . . . . . . 10 ⊢ (𝑓:𝑛–1-1-onto→𝐴 → 𝑓:𝑛⟶𝐴)
17	16	adantl 271	. . . . . . . . 9 ⊢ (((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) ∧ 𝑓:𝑛–1-1-onto→𝐴) → 𝑓:𝑛⟶𝐴)
18		sucidg 4210	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ω → 𝑚 ∈ suc 𝑚)
19	18	ad3antlr 477	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) ∧ 𝑓:𝑛–1-1-onto→𝐴) → 𝑚 ∈ suc 𝑚)
20		simplr 497	. . . . . . . . . 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 2164	. . . . . . . . 9 ⊢ (((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) ∧ 𝑓:𝑛–1-1-onto→𝐴) → 𝑚 ∈ 𝑛)
22	17, 21	ffvelrnd 5383	. . . . . . . 8 ⊢ (((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) ∧ 𝑓:𝑛–1-1-onto→𝐴) → (𝑓‘𝑚) ∈ 𝐴)
23		elex2 2628	. . . . . . . 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 1823	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → ∃𝑥 𝑥 ∈ 𝐴)
26	25	ex 113	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) → (𝑛 = suc 𝑚 → ∃𝑥 𝑥 ∈ 𝐴))
27	26	rexlimdva 2485	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → (∃𝑚 ∈ ω 𝑛 = suc 𝑚 → ∃𝑥 𝑥 ∈ 𝐴))
28	10, 27	orim12d 733	. . 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 2489	1 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ∃𝑥 𝑥 ∈ 𝐴))

Syntax hints:   → wi 4   ∧ wa 102   ∨ wo 662   = wceq 1287  ∃wex 1424   ∈ wcel 1436  ∃wrex 2356  ∅c0 3272   class class class wbr 3814  suc csuc 4159  ωcom 4371  ⟶wf 4968  –1-1-onto→wf1o 4971  ‘cfv 4972   ≈ cen 6388  Fincfn 6390

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-iinf 4369

This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-sbc 2829  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-br 3815  df-opab 3869  df-id 4087  df-suc 4165  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-er 6225  df-en 6391  df-fin 6393

This theorem is referenced by:  xpfi  6568