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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 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 6532 . . . 4 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
21biimpi 119 . . 3 (𝐴 ∈ Fin → ∃𝑥 ∈ ω 𝐴𝑥)
323ad2ant1 965 . 2 ((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) → ∃𝑥 ∈ ω 𝐴𝑥)
4 bren 6518 . . . . 5 (𝐴𝑥 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝑥)
54biimpi 119 . . . 4 (𝐴𝑥 → ∃𝑓 𝑓:𝐴1-1-onto𝑥)
65ad2antll 476 . . 3 (((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) → ∃𝑓 𝑓:𝐴1-1-onto𝑥)
7 f1of 5266 . . . . . . . . . 10 (𝑓:𝐴1-1-onto𝑥𝑓:𝐴𝑥)
87adantl 272 . . . . . . . . 9 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝑓:𝐴𝑥)
9 simpll2 984 . . . . . . . . 9 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝐵𝐴)
108, 9ffvelrnd 5449 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (𝑓𝐵) ∈ 𝑥)
11 simplrl 503 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝑥 ∈ ω)
12 elnn 4433 . . . . . . . 8 (((𝑓𝐵) ∈ 𝑥𝑥 ∈ ω) → (𝑓𝐵) ∈ ω)
1310, 11, 12syl2anc 404 . . . . . . 7 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (𝑓𝐵) ∈ ω)
14 simpll3 985 . . . . . . . . 9 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝐶𝐴)
158, 14ffvelrnd 5449 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (𝑓𝐶) ∈ 𝑥)
16 elnn 4433 . . . . . . . 8 (((𝑓𝐶) ∈ 𝑥𝑥 ∈ ω) → (𝑓𝐶) ∈ ω)
1715, 11, 16syl2anc 404 . . . . . . 7 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (𝑓𝐶) ∈ ω)
18 nndceq 6274 . . . . . . 7 (((𝑓𝐵) ∈ ω ∧ (𝑓𝐶) ∈ ω) → DECID (𝑓𝐵) = (𝑓𝐶))
1913, 17, 18syl2anc 404 . . . . . 6 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → DECID (𝑓𝐵) = (𝑓𝐶))
20 exmiddc 783 . . . . . 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 5265 . . . . . . . 8 (𝑓:𝐴1-1-onto𝑥𝑓:𝐴1-1𝑥)
2322adantl 272 . . . . . . 7 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝑓:𝐴1-1𝑥)
24 f1veqaeq 5562 . . . . . . 7 ((𝑓:𝐴1-1𝑥 ∧ (𝐵𝐴𝐶𝐴)) → ((𝑓𝐵) = (𝑓𝐶) → 𝐵 = 𝐶))
2523, 9, 14, 24syl12anc 1173 . . . . . 6 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → ((𝑓𝐵) = (𝑓𝐶) → 𝐵 = 𝐶))
26 fveq2 5318 . . . . . . . 8 (𝐵 = 𝐶 → (𝑓𝐵) = (𝑓𝐶))
2726con3i 598 . . . . . . 7 (¬ (𝑓𝐵) = (𝑓𝐶) → ¬ 𝐵 = 𝐶)
2827a1i 9 . . . . . 6 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (¬ (𝑓𝐵) = (𝑓𝐶) → ¬ 𝐵 = 𝐶))
2925, 28orim12d 736 . . . . 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 782 . . . 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 1827 . 2 (((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) → DECID 𝐵 = 𝐶)
343, 33rexlimddv 2494 1 ((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) → DECID 𝐵 = 𝐶)
Colors of variables: wff set class
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  1-1wf1 5025  1-1-ontowf1o 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
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