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Theorem fidceq 6871
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 6763 . . . 4 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
21biimpi 120 . . 3 (𝐴 ∈ Fin → ∃𝑥 ∈ ω 𝐴𝑥)
323ad2ant1 1018 . 2 ((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) → ∃𝑥 ∈ ω 𝐴𝑥)
4 bren 6749 . . . . 5 (𝐴𝑥 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝑥)
54biimpi 120 . . . 4 (𝐴𝑥 → ∃𝑓 𝑓:𝐴1-1-onto𝑥)
65ad2antll 491 . . 3 (((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) → ∃𝑓 𝑓:𝐴1-1-onto𝑥)
7 f1of 5463 . . . . . . . . . 10 (𝑓:𝐴1-1-onto𝑥𝑓:𝐴𝑥)
87adantl 277 . . . . . . . . 9 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝑓:𝐴𝑥)
9 simpll2 1037 . . . . . . . . 9 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝐵𝐴)
108, 9ffvelcdmd 5654 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (𝑓𝐵) ∈ 𝑥)
11 simplrl 535 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝑥 ∈ ω)
12 elnn 4607 . . . . . . . 8 (((𝑓𝐵) ∈ 𝑥𝑥 ∈ ω) → (𝑓𝐵) ∈ ω)
1310, 11, 12syl2anc 411 . . . . . . 7 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (𝑓𝐵) ∈ ω)
14 simpll3 1038 . . . . . . . . 9 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝐶𝐴)
158, 14ffvelcdmd 5654 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (𝑓𝐶) ∈ 𝑥)
16 elnn 4607 . . . . . . . 8 (((𝑓𝐶) ∈ 𝑥𝑥 ∈ ω) → (𝑓𝐶) ∈ ω)
1715, 11, 16syl2anc 411 . . . . . . 7 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (𝑓𝐶) ∈ ω)
18 nndceq 6502 . . . . . . 7 (((𝑓𝐵) ∈ ω ∧ (𝑓𝐶) ∈ ω) → DECID (𝑓𝐵) = (𝑓𝐶))
1913, 17, 18syl2anc 411 . . . . . 6 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → DECID (𝑓𝐵) = (𝑓𝐶))
20 exmiddc 836 . . . . . 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 5462 . . . . . . . 8 (𝑓:𝐴1-1-onto𝑥𝑓:𝐴1-1𝑥)
2322adantl 277 . . . . . . 7 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → 𝑓:𝐴1-1𝑥)
24 f1veqaeq 5772 . . . . . . 7 ((𝑓:𝐴1-1𝑥 ∧ (𝐵𝐴𝐶𝐴)) → ((𝑓𝐵) = (𝑓𝐶) → 𝐵 = 𝐶))
2523, 9, 14, 24syl12anc 1236 . . . . . 6 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → ((𝑓𝐵) = (𝑓𝐶) → 𝐵 = 𝐶))
26 fveq2 5517 . . . . . . . 8 (𝐵 = 𝐶 → (𝑓𝐵) = (𝑓𝐶))
2726con3i 632 . . . . . . 7 (¬ (𝑓𝐵) = (𝑓𝐶) → ¬ 𝐵 = 𝐶)
2827a1i 9 . . . . . 6 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → (¬ (𝑓𝐵) = (𝑓𝐶) → ¬ 𝐵 = 𝐶))
2925, 28orim12d 786 . . . . 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 835 . . . 4 (DECID 𝐵 = 𝐶 ↔ (𝐵 = 𝐶 ∨ ¬ 𝐵 = 𝐶))
3230, 31sylibr 134 . . 3 ((((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) ∧ 𝑓:𝐴1-1-onto𝑥) → DECID 𝐵 = 𝐶)
336, 32exlimddv 1898 . 2 (((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) ∧ (𝑥 ∈ ω ∧ 𝐴𝑥)) → DECID 𝐵 = 𝐶)
343, 33rexlimddv 2599 1 ((𝐴 ∈ Fin ∧ 𝐵𝐴𝐶𝐴) → DECID 𝐵 = 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 708  DECID wdc 834  w3a 978   = wceq 1353  wex 1492  wcel 2148  wrex 2456   class class class wbr 4005  ωcom 4591  wf 5214  1-1wf1 5215  1-1-ontowf1o 5217  cfv 5218  cen 6740  Fincfn 6742
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-en 6743  df-fin 6745
This theorem is referenced by:  fidifsnen  6872  fidifsnid  6873  pw1fin  6912  unfiexmid  6919  undiffi  6926  fidcenumlemim  6953
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