Theorem fidceq 7126

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 7002	. . . 4 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
2	1	biimpi 120	. . 3 ⊢ (𝐴 ∈ Fin → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
3	2	3ad2ant1 1045	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
4		bren 6985	. . . . 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 5616	. . . . . . . . . 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 1064	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → 𝐵 ∈ 𝐴)
10	8, 9	ffvelcdmd 5815	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → (𝑓‘𝐵) ∈ 𝑥)
11		simplrl 537	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → 𝑥 ∈ ω)
12		elnn 4730	. . . . . . . 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 1065	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → 𝐶 ∈ 𝐴)
15	8, 14	ffvelcdmd 5815	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) ∧ 𝑓:𝐴–1-1-onto→𝑥) → (𝑓‘𝐶) ∈ 𝑥)
16		elnn 4730	. . . . . . . 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 6734	. . . . . . 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 844	. . . . . 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 5615	. . . . . . . 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 5944	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1→𝑥 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ((𝑓‘𝐵) = (𝑓‘𝐶) → 𝐵 = 𝐶))
25	23, 9, 14, 24	syl12anc 1272	. . . . . 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 5672	. . . . . . . 8 ⊢ (𝐵 = 𝐶 → (𝑓‘𝐵) = (𝑓‘𝐶))
27	26	con3i 637	. . . . . . 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 794	. . . . 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 843	. . . 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 1950	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴 ≈ 𝑥)) → DECID 𝐵 = 𝐶)
34	3, 33	rexlimddv 2667	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → DECID 𝐵 = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 716  DECID wdc 842   ∧ w3a 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2205  ∃wrex 2523   class class class wbr 4111  ωcom 4714  ⟶wf 5350  –1-1→wf1 5351  –1-1-onto→wf1o 5353  ‘cfv 5354   ≈ cen 6975  Fincfn 6977

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-en 6978  df-fin 6980

This theorem is referenced by:  fidifsnen  7127  fidifsnid  7128  pw1fin  7172  unfiexmid  7180  undiffi  7187  fissfi  7218  fidcenumlemim  7224  ballotfilem2  13153  vtxedgfi  16333  vtxlpfi  16334  eupth2lem3lem3fi  16514  eupth2lem3lem4fi  16517  eupth2lem3lem7fi  16518  eupth2lemsfi  16522  eulerpathprum  16524