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Theorem fidceq 6360
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 6271 . . . 4 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
21biimpi 117 . . 3 (𝐴 ∈ Fin → ∃𝑥 ∈ ω 𝐴𝑥)
323ad2ant1 936 . 2 ((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) → ∃𝑥 ∈ ω 𝐴𝑥)
4 bren 6258 . . . . 5 (𝐴𝑥 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝑥)
54biimpi 117 . . . 4 (𝐴𝑥 → ∃𝑓 𝑓:𝐴1-1-onto𝑥)
65ad2antll 468 . . 3 (((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) → ∃𝑓 𝑓:𝐴1-1-onto𝑥)
7 f1of 5153 . . . . . . . . . 10 (𝑓:𝐴1-1-onto𝑥𝑓:𝐴𝑥)
87adantl 266 . . . . . . . . 9 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝑓:𝐴𝑥)
9 simpll2 955 . . . . . . . . 9 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝐵𝐴)
108, 9ffvelrnd 5330 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (𝑓𝐵) ∈ 𝑥)
11 simplrl 495 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝑥 ∈ ω)
12 elnn 4355 . . . . . . . 8 (((𝑓𝐵) ∈ 𝑥𝑥 ∈ ω) → (𝑓𝐵) ∈ ω)
1310, 11, 12syl2anc 397 . . . . . . 7 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (𝑓𝐵) ∈ ω)
14 simpll3 956 . . . . . . . . 9 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝐶𝐴)
158, 14ffvelrnd 5330 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (𝑓𝐶) ∈ 𝑥)
16 elnn 4355 . . . . . . . 8 (((𝑓𝐶) ∈ 𝑥𝑥 ∈ ω) → (𝑓𝐶) ∈ ω)
1715, 11, 16syl2anc 397 . . . . . . 7 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (𝑓𝐶) ∈ ω)
18 nndceq 6107 . . . . . . 7 (((𝑓𝐵) ∈ ω ∧ (𝑓𝐶) ∈ ω) → DECID (𝑓𝐵) = (𝑓𝐶))
1913, 17, 18syl2anc 397 . . . . . 6 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → DECID (𝑓𝐵) = (𝑓𝐶))
20 exmiddc 755 . . . . . 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 5152 . . . . . . . 8 (𝑓:𝐴1-1-onto𝑥𝑓:𝐴1-1𝑥)
2322adantl 266 . . . . . . 7 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝑓:𝐴1-1𝑥)
24 f1veqaeq 5435 . . . . . . 7 ((𝑓:𝐴1-1𝑥 ∧ (𝐵𝐴𝐶𝐴)) → ((𝑓𝐵) = (𝑓𝐶) → 𝐵 = 𝐶))
2523, 9, 14, 24syl12anc 1144 . . . . . 6 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → ((𝑓𝐵) = (𝑓𝐶) → 𝐵 = 𝐶))
26 fveq2 5205 . . . . . . . 8 (𝐵 = 𝐶 → (𝑓𝐵) = (𝑓𝐶))
2726con3i 572 . . . . . . 7 (¬ (𝑓𝐵) = (𝑓𝐶) → ¬ 𝐵 = 𝐶)
2827a1i 9 . . . . . 6 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (¬ (𝑓𝐵) = (𝑓𝐶) → ¬ 𝐵 = 𝐶))
2925, 28orim12d 710 . . . . 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 754 . . . 4 (DECID 𝐵 = 𝐶 ↔ (𝐵 = 𝐶 ∨ ¬ 𝐵 = 𝐶))
3230, 31sylibr 141 . . 3 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → DECID 𝐵 = 𝐶)
336, 32exlimddv 1794 . 2 (((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) → DECID 𝐵 = 𝐶)
343, 33rexlimddv 2454 1 ((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) → DECID 𝐵 = 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wo 639  DECID wdc 753  w3a 896   = wceq 1259  wex 1397  wcel 1409  wrex 2324   class class class wbr 3791  ωcom 4340  wf 4925  1-1wf1 4926  1-1-ontowf1o 4928  cfv 4929  cen 6249  Fincfn 6251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-br 3792  df-opab 3846  df-tr 3882  df-id 4057  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-en 6252  df-fin 6254
This theorem is referenced by:  fidifsnen  6361  fidifsnid  6362
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