Theorem fidceq 6639

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 6532	. . . 4 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
2	1	biimpi 119	. . 3 ⊢ (𝐴 ∈ Fin → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
3	2	3ad2ant1 965	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
4		bren 6518	. . . . 5 ⊢ (𝐴 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝑥)
5	4	biimpi 119	. . . 4 ⊢ (𝐴 ≈ 𝑥 → ∃𝑓 𝑓:𝐴–1-1-onto→𝑥)
6	5	ad2antll 476	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) → ∃𝑓 𝑓:𝐴–1-1-onto→𝑥)
7		f1of 5266	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓:𝐴⟶𝑥)
8	7	adantl 272	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → 𝑓:𝐴⟶𝑥)
9		simpll2 984	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → 𝐵 ∈ 𝐴)
10	8, 9	ffvelrnd 5449	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → (𝑓‘𝐵) ∈ 𝑥)
11		simplrl 503	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → 𝑥 ∈ ω)
12		elnn 4433	. . . . . . . 8 ⊢ (((𝑓‘𝐵) ∈ 𝑥 ∧ 𝑥 ∈ ω) → (𝑓‘𝐵) ∈ ω)
13	10, 11, 12	syl2anc 404	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → (𝑓‘𝐵) ∈ ω)
14		simpll3 985	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → 𝐶 ∈ 𝐴)
15	8, 14	ffvelrnd 5449	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → (𝑓‘𝐶) ∈ 𝑥)
16		elnn 4433	. . . . . . . 8 ⊢ (((𝑓‘𝐶) ∈ 𝑥 ∧ 𝑥 ∈ ω) → (𝑓‘𝐶) ∈ ω)
17	15, 11, 16	syl2anc 404	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → (𝑓‘𝐶) ∈ ω)
18		nndceq 6274	. . . . . . 7 ⊢ (((𝑓‘𝐵) ∈ ω ∧ (𝑓‘𝐶) ∈ ω) → DECID (𝑓‘𝐵) = (𝑓‘𝐶))
19	13, 17, 18	syl2anc 404	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → DECID (𝑓‘𝐵) = (𝑓‘𝐶))
20		exmiddc 783	. . . . . 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 5265	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓:𝐴–1-1→𝑥)
23	22	adantl 272	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → 𝑓:𝐴–1-1→𝑥)
24		f1veqaeq 5562	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1→𝑥 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ((𝑓‘𝐵) = (𝑓‘𝐶) → 𝐵 = 𝐶))
25	23, 9, 14, 24	syl12anc 1173	. . . . . 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 5318	. . . . . . . 8 ⊢ (𝐵 = 𝐶 → (𝑓‘𝐵) = (𝑓‘𝐶))
27	26	con3i 598	. . . . . . 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 736	. . . . 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 782	. . . 4 ⊢ (DECID 𝐵 = 𝐶 ↔ (𝐵 = 𝐶 ∨ ¬ 𝐵 = 𝐶))
32	30, 31	sylibr 133	. . 3 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → DECID 𝐵 = 𝐶)
33	6, 32	exlimddv 1827	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) → DECID 𝐵 = 𝐶)
34	3, 33	rexlimddv 2494	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → DECID 𝐵 = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 665  DECID wdc 781   ∧ w3a 925   = wceq 1290  ∃wex 1427   ∈ wcel 1439  ∃wrex 2361   class class class wbr 3851  ωcom 4418  ⟶wf 5024  –1-1→wf1 5025  –1-1-onto→wf1o 5027  ‘cfv 5028   ≈ cen 6509  Fincfn 6511

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-br 3852  df-opab 3906  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-en 6512  df-fin 6514

This theorem is referenced by:  fidifsnen  6640  fidifsnid  6641  unfiexmid  6682  undiffi  6689  fidcenumlemim  6715