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Theorem fidceq 6731
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 1o or to 2o 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 ∧ 𝐵𝐴𝐶𝐴) → DECID 𝐵 = 𝐶)

Proof of Theorem fidceq
Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isfi 6623 . . . 4 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
21biimpi 119 . . 3 (𝐴 ∈ Fin → ∃𝑥 ∈ ω 𝐴𝑥)
323ad2ant1 987 . 2 ((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) → ∃𝑥 ∈ ω 𝐴𝑥)
4 bren 6609 . . . . 5 (𝐴𝑥 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝑥)
54biimpi 119 . . . 4 (𝐴𝑥 → ∃𝑓 𝑓:𝐴1-1-onto𝑥)
65ad2antll 482 . . 3 (((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) → ∃𝑓 𝑓:𝐴1-1-onto𝑥)
7 f1of 5335 . . . . . . . . . 10 (𝑓:𝐴1-1-onto𝑥𝑓:𝐴𝑥)
87adantl 275 . . . . . . . . 9 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝑓:𝐴𝑥)
9 simpll2 1006 . . . . . . . . 9 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝐵𝐴)
108, 9ffvelrnd 5524 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (𝑓𝐵) ∈ 𝑥)
11 simplrl 509 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝑥 ∈ ω)
12 elnn 4489 . . . . . . . 8 (((𝑓𝐵) ∈ 𝑥𝑥 ∈ ω) → (𝑓𝐵) ∈ ω)
1310, 11, 12syl2anc 408 . . . . . . 7 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (𝑓𝐵) ∈ ω)
14 simpll3 1007 . . . . . . . . 9 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝐶𝐴)
158, 14ffvelrnd 5524 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (𝑓𝐶) ∈ 𝑥)
16 elnn 4489 . . . . . . . 8 (((𝑓𝐶) ∈ 𝑥𝑥 ∈ ω) → (𝑓𝐶) ∈ ω)
1715, 11, 16syl2anc 408 . . . . . . 7 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (𝑓𝐶) ∈ ω)
18 nndceq 6363 . . . . . . 7 (((𝑓𝐵) ∈ ω ∧ (𝑓𝐶) ∈ ω) → DECID (𝑓𝐵) = (𝑓𝐶))
1913, 17, 18syl2anc 408 . . . . . 6 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → DECID (𝑓𝐵) = (𝑓𝐶))
20 exmiddc 806 . . . . . 6 (DECID (𝑓𝐵) = (𝑓𝐶) → ((𝑓𝐵) = (𝑓𝐶) ∨ ¬ (𝑓𝐵) = (𝑓𝐶)))
2119, 20syl 14 . . . . 5 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → ((𝑓𝐵) = (𝑓𝐶) ∨ ¬ (𝑓𝐵) = (𝑓𝐶)))
22 f1of1 5334 . . . . . . . 8 (𝑓:𝐴1-1-onto𝑥𝑓:𝐴1-1𝑥)
2322adantl 275 . . . . . . 7 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝑓:𝐴1-1𝑥)
24 f1veqaeq 5638 . . . . . . 7 ((𝑓:𝐴1-1𝑥 ∧ (𝐵𝐴𝐶𝐴)) → ((𝑓𝐵) = (𝑓𝐶) → 𝐵 = 𝐶))
2523, 9, 14, 24syl12anc 1199 . . . . . 6 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → ((𝑓𝐵) = (𝑓𝐶) → 𝐵 = 𝐶))
26 fveq2 5389 . . . . . . . 8 (𝐵 = 𝐶 → (𝑓𝐵) = (𝑓𝐶))
2726con3i 606 . . . . . . 7 (¬ (𝑓𝐵) = (𝑓𝐶) → ¬ 𝐵 = 𝐶)
2827a1i 9 . . . . . 6 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (¬ (𝑓𝐵) = (𝑓𝐶) → ¬ 𝐵 = 𝐶))
2925, 28orim12d 760 . . . . 5 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (((𝑓𝐵) = (𝑓𝐶) ∨ ¬ (𝑓𝐵) = (𝑓𝐶)) → (𝐵 = 𝐶 ∨ ¬ 𝐵 = 𝐶)))
3021, 29mpd 13 . . . 4 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (𝐵 = 𝐶 ∨ ¬ 𝐵 = 𝐶))
31 df-dc 805 . . . 4 (DECID 𝐵 = 𝐶 ↔ (𝐵 = 𝐶 ∨ ¬ 𝐵 = 𝐶))
3230, 31sylibr 133 . . 3 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → DECID 𝐵 = 𝐶)
336, 32exlimddv 1854 . 2 (((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) → DECID 𝐵 = 𝐶)
343, 33rexlimddv 2531 1 ((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) → DECID 𝐵 = 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 682  DECID wdc 804  w3a 947   = wceq 1316  wex 1453  wcel 1465  wrex 2394   class class class wbr 3899  ωcom 4474  wf 5089  1-1wf1 5090  1-1-ontowf1o 5092  cfv 5093  cen 6600  Fincfn 6602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-en 6603  df-fin 6605
This theorem is referenced by:  fidifsnen  6732  fidifsnid  6733  unfiexmid  6774  undiffi  6781  fidcenumlemim  6808
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