Theorem fidceq 6973

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 6859	. . . 4 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
2	1	biimpi 120	. . 3 ⊢ (𝐴 ∈ Fin → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
3	2	3ad2ant1 1021	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
4		bren 6842	. . . . 5 ⊢ (𝐴 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝑥)
5	4	biimpi 120	. . . 4 ⊢ (𝐴 ≈ 𝑥 → ∃𝑓 𝑓:𝐴–1-1-onto→𝑥)
6	5	ad2antll 491	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) → ∃𝑓 𝑓:𝐴–1-1-onto→𝑥)
7		f1of 5529	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓:𝐴⟶𝑥)
8	7	adantl 277	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → 𝑓:𝐴⟶𝑥)
9		simpll2 1040	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → 𝐵 ∈ 𝐴)
10	8, 9	ffvelcdmd 5723	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → (𝑓‘𝐵) ∈ 𝑥)
11		simplrl 535	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → 𝑥 ∈ ω)
12		elnn 4658	. . . . . . . 8 ⊢ (((𝑓‘𝐵) ∈ 𝑥 ∧ 𝑥 ∈ ω) → (𝑓‘𝐵) ∈ ω)
13	10, 11, 12	syl2anc 411	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → (𝑓‘𝐵) ∈ ω)
14		simpll3 1041	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → 𝐶 ∈ 𝐴)
15	8, 14	ffvelcdmd 5723	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → (𝑓‘𝐶) ∈ 𝑥)
16		elnn 4658	. . . . . . . 8 ⊢ (((𝑓‘𝐶) ∈ 𝑥 ∧ 𝑥 ∈ ω) → (𝑓‘𝐶) ∈ ω)
17	15, 11, 16	syl2anc 411	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → (𝑓‘𝐶) ∈ ω)
18		nndceq 6592	. . . . . . 7 ⊢ (((𝑓‘𝐵) ∈ ω ∧ (𝑓‘𝐶) ∈ ω) → DECID (𝑓‘𝐵) = (𝑓‘𝐶))
19	13, 17, 18	syl2anc 411	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → DECID (𝑓‘𝐵) = (𝑓‘𝐶))
20		exmiddc 838	. . . . . 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 5528	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→𝑥 → 𝑓:𝐴–1-1→𝑥)
23	22	adantl 277	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → 𝑓:𝐴–1-1→𝑥)
24		f1veqaeq 5845	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1→𝑥 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ((𝑓‘𝐵) = (𝑓‘𝐶) → 𝐵 = 𝐶))
25	23, 9, 14, 24	syl12anc 1248	. . . . . 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 5583	. . . . . . . 8 ⊢ (𝐵 = 𝐶 → (𝑓‘𝐵) = (𝑓‘𝐶))
27	26	con3i 633	. . . . . . 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 788	. . . . 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 837	. . . 4 ⊢ (DECID 𝐵 = 𝐶 ↔ (𝐵 = 𝐶 ∨ ¬ 𝐵 = 𝐶))
32	30, 31	sylibr 134	. . 3 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → DECID 𝐵 = 𝐶)
33	6, 32	exlimddv 1923	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) → DECID 𝐵 = 𝐶)
34	3, 33	rexlimddv 2629	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → DECID 𝐵 = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 710  DECID wdc 836   ∧ w3a 981   = wceq 1373  ∃wex 1516   ∈ wcel 2177  ∃wrex 2486   class class class wbr 4047  ωcom 4642  ⟶wf 5272  –1-1→wf1 5273  –1-1-onto→wf1o 5275  ‘cfv 5276   ≈ cen 6832  Fincfn 6834

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3000  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-br 4048  df-opab 4110  df-tr 4147  df-id 4344  df-iord 4417  df-on 4419  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-en 6835  df-fin 6837

This theorem is referenced by:  fidifsnen  6974  fidifsnid  6975  pw1fin  7014  unfiexmid  7022  undiffi  7029  fidcenumlemim  7061