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Theorem fidceq 6537
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 6430 . . . 4 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
21biimpi 118 . . 3 (𝐴 ∈ Fin → ∃𝑥 ∈ ω 𝐴𝑥)
323ad2ant1 962 . 2 ((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) → ∃𝑥 ∈ ω 𝐴𝑥)
4 bren 6416 . . . . 5 (𝐴𝑥 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝑥)
54biimpi 118 . . . 4 (𝐴𝑥 → ∃𝑓 𝑓:𝐴1-1-onto𝑥)
65ad2antll 475 . . 3 (((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) → ∃𝑓 𝑓:𝐴1-1-onto𝑥)
7 f1of 5216 . . . . . . . . . 10 (𝑓:𝐴1-1-onto𝑥𝑓:𝐴𝑥)
87adantl 271 . . . . . . . . 9 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝑓:𝐴𝑥)
9 simpll2 981 . . . . . . . . 9 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝐵𝐴)
108, 9ffvelrnd 5398 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (𝑓𝐵) ∈ 𝑥)
11 simplrl 502 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝑥 ∈ ω)
12 elnn 4393 . . . . . . . 8 (((𝑓𝐵) ∈ 𝑥𝑥 ∈ ω) → (𝑓𝐵) ∈ ω)
1310, 11, 12syl2anc 403 . . . . . . 7 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (𝑓𝐵) ∈ ω)
14 simpll3 982 . . . . . . . . 9 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝐶𝐴)
158, 14ffvelrnd 5398 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (𝑓𝐶) ∈ 𝑥)
16 elnn 4393 . . . . . . . 8 (((𝑓𝐶) ∈ 𝑥𝑥 ∈ ω) → (𝑓𝐶) ∈ ω)
1715, 11, 16syl2anc 403 . . . . . . 7 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (𝑓𝐶) ∈ ω)
18 nndceq 6214 . . . . . . 7 (((𝑓𝐵) ∈ ω ∧ (𝑓𝐶) ∈ ω) → DECID (𝑓𝐵) = (𝑓𝐶))
1913, 17, 18syl2anc 403 . . . . . 6 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → DECID (𝑓𝐵) = (𝑓𝐶))
20 exmiddc 780 . . . . . 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 5215 . . . . . . . 8 (𝑓:𝐴1-1-onto𝑥𝑓:𝐴1-1𝑥)
2322adantl 271 . . . . . . 7 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝑓:𝐴1-1𝑥)
24 f1veqaeq 5509 . . . . . . 7 ((𝑓:𝐴1-1𝑥 ∧ (𝐵𝐴𝐶𝐴)) → ((𝑓𝐵) = (𝑓𝐶) → 𝐵 = 𝐶))
2523, 9, 14, 24syl12anc 1170 . . . . . 6 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → ((𝑓𝐵) = (𝑓𝐶) → 𝐵 = 𝐶))
26 fveq2 5268 . . . . . . . 8 (𝐵 = 𝐶 → (𝑓𝐵) = (𝑓𝐶))
2726con3i 595 . . . . . . 7 (¬ (𝑓𝐵) = (𝑓𝐶) → ¬ 𝐵 = 𝐶)
2827a1i 9 . . . . . 6 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (¬ (𝑓𝐵) = (𝑓𝐶) → ¬ 𝐵 = 𝐶))
2925, 28orim12d 733 . . . . 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 779 . . . 4 (DECID 𝐵 = 𝐶 ↔ (𝐵 = 𝐶 ∨ ¬ 𝐵 = 𝐶))
3230, 31sylibr 132 . . 3 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → DECID 𝐵 = 𝐶)
336, 32exlimddv 1823 . 2 (((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) → DECID 𝐵 = 𝐶)
343, 33rexlimddv 2489 1 ((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) → DECID 𝐵 = 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 662  DECID wdc 778  w3a 922   = wceq 1287  wex 1424  wcel 1436  wrex 2356   class class class wbr 3820  ωcom 4378  wf 4977  1-1wf1 4978  1-1-ontowf1o 4980  cfv 4981  cen 6407  Fincfn 6409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-nul 3940  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-iinf 4376
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-br 3821  df-opab 3875  df-tr 3912  df-id 4094  df-iord 4167  df-on 4169  df-suc 4172  df-iom 4379  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989  df-en 6410  df-fin 6412
This theorem is referenced by:  fidifsnen  6538  fidifsnid  6539  unfiexmid  6580  undiffi  6587
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