Theorem fidceq 6330

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 6241	. . . 4 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
2	1	biimpi 113	. . 3 ⊢ (𝐴 ∈ Fin → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
3	2	3ad2ant1 925	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
4		bren 6228	. . . . 5 ⊢ (𝐴 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝑥)
5	4	biimpi 113	. . . 4 ⊢ (𝐴 ≈ 𝑥 → ∃𝑓 𝑓:𝐴–1-1-onto→𝑥)
6	5	ad2antll 460	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) → ∃𝑓 𝑓:𝐴–1-1-onto→𝑥)
7		f1of 5126	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓:𝐴⟶𝑥)
8	7	adantl 262	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → 𝑓:𝐴⟶𝑥)
9		simpll2 944	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → 𝐵 ∈ 𝐴)
10	8, 9	ffvelrnd 5303	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → (𝑓‘𝐵) ∈ 𝑥)
11		simplrl 487	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → 𝑥 ∈ ω)
12		elnn 4328	. . . . . . . 8 ⊢ (((𝑓‘𝐵) ∈ 𝑥 ∧ 𝑥 ∈ ω) → (𝑓‘𝐵) ∈ ω)
13	10, 11, 12	syl2anc 391	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → (𝑓‘𝐵) ∈ ω)
14		simpll3 945	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → 𝐶 ∈ 𝐴)
15	8, 14	ffvelrnd 5303	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → (𝑓‘𝐶) ∈ 𝑥)
16		elnn 4328	. . . . . . . 8 ⊢ (((𝑓‘𝐶) ∈ 𝑥 ∧ 𝑥 ∈ ω) → (𝑓‘𝐶) ∈ ω)
17	15, 11, 16	syl2anc 391	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → (𝑓‘𝐶) ∈ ω)
18		nndceq 6077	. . . . . . 7 ⊢ (((𝑓‘𝐵) ∈ ω ∧ (𝑓‘𝐶) ∈ ω) → DECID (𝑓‘𝐵) = (𝑓‘𝐶))
19	13, 17, 18	syl2anc 391	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → DECID (𝑓‘𝐵) = (𝑓‘𝐶))
20		exmiddc 744	. . . . . 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 5125	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓:𝐴–1-1→𝑥)
23	22	adantl 262	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → 𝑓:𝐴–1-1→𝑥)
24		f1veqaeq 5408	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1→𝑥 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ((𝑓‘𝐵) = (𝑓‘𝐶) → 𝐵 = 𝐶))
25	23, 9, 14, 24	syl12anc 1133	. . . . . 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 5178	. . . . . . . 8 ⊢ (𝐵 = 𝐶 → (𝑓‘𝐵) = (𝑓‘𝐶))
27	26	con3i 562	. . . . . . 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 700	. . . . 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 743	. . . 4 ⊢ (DECID 𝐵 = 𝐶 ↔ (𝐵 = 𝐶 ∨ ¬ 𝐵 = 𝐶))
32	30, 31	sylibr 137	. . 3 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → DECID 𝐵 = 𝐶)
33	6, 32	exlimddv 1778	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) → DECID 𝐵 = 𝐶)
34	3, 33	rexlimddv 2437	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → DECID 𝐵 = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∨ wo 629  DECID wdc 742   ∧ w3a 885   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃wrex 2307   class class class wbr 3764  ωcom 4313  ⟶wf 4898  –1-1→wf1 4899  –1-1-onto→wf1o 4901  ‘cfv 4902   ≈ cen 6219  Fincfn 6221

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-opab 3819  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-en 6222  df-fin 6224

This theorem is referenced by:  fidifsnen  6331  fidifsnid  6332