Theorem fidceq 6354

Description: Equality of members of a finite set is decidable. This may be counterintuitive: cannot any two sets be elements of a finite set? Well, to show, for example, that {𝐵, 𝐶} is finite would require showing it is equinumerous to 1_𝑜 or to 2_𝑜 but to show that you'd need to know 𝐵 = 𝐶 or ¬ 𝐵 = 𝐶, respectively. (Contributed by Jim Kingdon, 5-Sep-2021.)

Assertion

Ref	Expression
fidceq	⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → DECID 𝐵 = 𝐶)

Proof of Theorem fidceq

Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isfi 6264	. . . 4 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
2	1	biimpi 118	. . 3 ⊢ (𝐴 ∈ Fin → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
3	2	3ad2ant1 959	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
4		bren 6251	. . . . 5 ⊢ (𝐴 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝑥)
5	4	biimpi 118	. . . 4 ⊢ (𝐴 ≈ 𝑥 → ∃𝑓 𝑓:𝐴–1-1-onto→𝑥)
6	5	ad2antll 474	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) → ∃𝑓 𝑓:𝐴–1-1-onto→𝑥)
7		f1of 5146	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓:𝐴⟶𝑥)
8	7	adantl 271	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → 𝑓:𝐴⟶𝑥)
9		simpll2 978	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → 𝐵 ∈ 𝐴)
10	8, 9	ffvelrnd 5324	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → (𝑓‘𝐵) ∈ 𝑥)
11		simplrl 501	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → 𝑥 ∈ ω)
12		elnn 4346	. . . . . . . 8 ⊢ (((𝑓‘𝐵) ∈ 𝑥 ∧ 𝑥 ∈ ω) → (𝑓‘𝐵) ∈ ω)
13	10, 11, 12	syl2anc 403	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → (𝑓‘𝐵) ∈ ω)
14		simpll3 979	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → 𝐶 ∈ 𝐴)
15	8, 14	ffvelrnd 5324	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → (𝑓‘𝐶) ∈ 𝑥)
16		elnn 4346	. . . . . . . 8 ⊢ (((𝑓‘𝐶) ∈ 𝑥 ∧ 𝑥 ∈ ω) → (𝑓‘𝐶) ∈ ω)
17	15, 11, 16	syl2anc 403	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → (𝑓‘𝐶) ∈ ω)
18		nndceq 6100	. . . . . . 7 ⊢ (((𝑓‘𝐵) ∈ ω ∧ (𝑓‘𝐶) ∈ ω) → DECID (𝑓‘𝐵) = (𝑓‘𝐶))
19	13, 17, 18	syl2anc 403	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → DECID (𝑓‘𝐵) = (𝑓‘𝐶))
20		exmiddc 777	. . . . . 6 ⊢ (DECID (𝑓‘𝐵) = (𝑓‘𝐶) → ((𝑓‘𝐵) = (𝑓‘𝐶) ∨ ¬ (𝑓‘𝐵) = (𝑓‘𝐶)))
21	19, 20	syl 14	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → ((𝑓‘𝐵) = (𝑓‘𝐶) ∨ ¬ (𝑓‘𝐵) = (𝑓‘𝐶)))
22		f1of1 5145	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓:𝐴–1-1→𝑥)
23	22	adantl 271	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → 𝑓:𝐴–1-1→𝑥)
24		f1veqaeq 5429	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1→𝑥 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ((𝑓‘𝐵) = (𝑓‘𝐶) → 𝐵 = 𝐶))
25	23, 9, 14, 24	syl12anc 1167	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → ((𝑓‘𝐵) = (𝑓‘𝐶) → 𝐵 = 𝐶))
26		fveq2 5198	. . . . . . . 8 ⊢ (𝐵 = 𝐶 → (𝑓‘𝐵) = (𝑓‘𝐶))
27	26	con3i 594	. . . . . . 7 ⊢ (¬ (𝑓‘𝐵) = (𝑓‘𝐶) → ¬ 𝐵 = 𝐶)
28	27	a1i 9	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → (¬ (𝑓‘𝐵) = (𝑓‘𝐶) → ¬ 𝐵 = 𝐶))
29	25, 28	orim12d 732	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → (((𝑓‘𝐵) = (𝑓‘𝐶) ∨ ¬ (𝑓‘𝐵) = (𝑓‘𝐶)) → (𝐵 = 𝐶 ∨ ¬ 𝐵 = 𝐶)))
30	21, 29	mpd 13	. . . 4 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → (𝐵 = 𝐶 ∨ ¬ 𝐵 = 𝐶))
31		df-dc 776	. . . 4 ⊢ (DECID 𝐵 = 𝐶 ↔ (𝐵 = 𝐶 ∨ ¬ 𝐵 = 𝐶))
32	30, 31	sylibr 132	. . 3 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → DECID 𝐵 = 𝐶)
33	6, 32	exlimddv 1819	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) → DECID 𝐵 = 𝐶)
34	3, 33	rexlimddv 2481	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → DECID 𝐵 = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ∨ wo 661  DECID wdc 775   ∧ w3a 919   = wceq 1284  ∃wex 1421   ∈ wcel 1433  ∃wrex 2349   class class class wbr 3785  ωcom 4331  ⟶wf 4918  –1-1→wf1 4919  –1-1-onto→wf1o 4921  ‘cfv 4922   ≈ cen 6242  Fincfn 6244

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-opab 3840  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-en 6245  df-fin 6247

This theorem is referenced by:  fidifsnen  6355  fidifsnid  6356