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Theorem diffitest 6533
Description: If subtracting any set from a finite set gives a finite set, any proposition of the form ¬ 𝜑 is decidable. This is not a proof of full excluded middle, but it is close enough to show we won't be able to prove 𝐴 ∈ Fin → (𝐴𝐵) ∈ Fin. (Contributed by Jim Kingdon, 8-Sep-2021.)
Hypothesis
Ref Expression
diffitest.1 𝑎 ∈ Fin ∀𝑏(𝑎𝑏) ∈ Fin
Assertion
Ref Expression
diffitest 𝜑 ∨ ¬ ¬ 𝜑)
Distinct variable groups:   𝑎,𝑏   𝜑,𝑏
Allowed substitution hint:   𝜑(𝑎)

Proof of Theorem diffitest
Dummy variables 𝑥 𝑛 𝑤 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 3931 . . . . . 6 ∅ ∈ V
2 snfig 6461 . . . . . 6 (∅ ∈ V → {∅} ∈ Fin)
31, 2ax-mp 7 . . . . 5 {∅} ∈ Fin
4 diffitest.1 . . . . 5 𝑎 ∈ Fin ∀𝑏(𝑎𝑏) ∈ Fin
5 difeq1 3095 . . . . . . . 8 (𝑎 = {∅} → (𝑎𝑏) = ({∅} ∖ 𝑏))
65eleq1d 2151 . . . . . . 7 (𝑎 = {∅} → ((𝑎𝑏) ∈ Fin ↔ ({∅} ∖ 𝑏) ∈ Fin))
76albidv 1747 . . . . . 6 (𝑎 = {∅} → (∀𝑏(𝑎𝑏) ∈ Fin ↔ ∀𝑏({∅} ∖ 𝑏) ∈ Fin))
87rspcv 2708 . . . . 5 ({∅} ∈ Fin → (∀𝑎 ∈ Fin ∀𝑏(𝑎𝑏) ∈ Fin → ∀𝑏({∅} ∖ 𝑏) ∈ Fin))
93, 4, 8mp2 16 . . . 4 𝑏({∅} ∖ 𝑏) ∈ Fin
10 rabexg 3947 . . . . . 6 ({∅} ∈ Fin → {𝑥 ∈ {∅} ∣ 𝜑} ∈ V)
113, 10ax-mp 7 . . . . 5 {𝑥 ∈ {∅} ∣ 𝜑} ∈ V
12 difeq2 3096 . . . . . 6 (𝑏 = {𝑥 ∈ {∅} ∣ 𝜑} → ({∅} ∖ 𝑏) = ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
1312eleq1d 2151 . . . . 5 (𝑏 = {𝑥 ∈ {∅} ∣ 𝜑} → (({∅} ∖ 𝑏) ∈ Fin ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin))
1411, 13spcv 2702 . . . 4 (∀𝑏({∅} ∖ 𝑏) ∈ Fin → ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin)
159, 14ax-mp 7 . . 3 ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin
16 isfi 6408 . . 3 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin ↔ ∃𝑛 ∈ ω ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛)
1715, 16mpbi 143 . 2 𝑛 ∈ ω ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛
18 0elnn 4395 . . . . 5 (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
19 breq2 3815 . . . . . . . . . 10 (𝑛 = ∅ → (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ ∅))
20 en0 6442 . . . . . . . . . 10 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ ∅ ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅)
2119, 20syl6bb 194 . . . . . . . . 9 (𝑛 = ∅ → (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅))
2221biimpac 292 . . . . . . . 8 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 = ∅) → ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅)
23 rabeq0 3295 . . . . . . . . 9 ({𝑥 ∈ {∅} ∣ ¬ 𝜑} = ∅ ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑)
24 notrab 3259 . . . . . . . . . 10 ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = {𝑥 ∈ {∅} ∣ ¬ 𝜑}
2524eqeq1i 2090 . . . . . . . . 9 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅ ↔ {𝑥 ∈ {∅} ∣ ¬ 𝜑} = ∅)
261snm 3534 . . . . . . . . . 10 𝑤 𝑤 ∈ {∅}
27 r19.3rmv 3353 . . . . . . . . . 10 (∃𝑤 𝑤 ∈ {∅} → (¬ ¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑))
2826, 27ax-mp 7 . . . . . . . . 9 (¬ ¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑)
2923, 25, 283bitr4i 210 . . . . . . . 8 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅ ↔ ¬ ¬ 𝜑)
3022, 29sylib 120 . . . . . . 7 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 = ∅) → ¬ ¬ 𝜑)
3130olcd 686 . . . . . 6 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 = ∅) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
32 ensym 6428 . . . . . . . 8 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 ≈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
33 elex2 2626 . . . . . . . 8 (∅ ∈ 𝑛 → ∃𝑤 𝑤𝑛)
34 enm 6466 . . . . . . . 8 ((𝑛 ≈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∧ ∃𝑤 𝑤𝑛) → ∃𝑦 𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
3532, 33, 34syl2an 283 . . . . . . 7 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ ∅ ∈ 𝑛) → ∃𝑦 𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
36 biidd 170 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜑))
3736elrab 2759 . . . . . . . . . . 11 (𝑦 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ↔ (𝑦 ∈ {∅} ∧ ¬ 𝜑))
3837simprbi 269 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → ¬ 𝜑)
3938orcd 685 . . . . . . . . 9 (𝑦 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4039, 24eleq2s 2177 . . . . . . . 8 (𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4140exlimiv 1530 . . . . . . 7 (∃𝑦 𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4235, 41syl 14 . . . . . 6 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ ∅ ∈ 𝑛) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4331, 42jaodan 744 . . . . 5 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ (𝑛 = ∅ ∨ ∅ ∈ 𝑛)) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4418, 43sylan2 280 . . . 4 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 ∈ ω) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4544ancoms 264 . . 3 ((𝑛 ∈ ω ∧ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4645rexlimiva 2478 . 2 (∃𝑛 ∈ ω ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4717, 46ax-mp 7 1 𝜑 ∨ ¬ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 102  wb 103  wo 662  wal 1283   = wceq 1285  wex 1422  wcel 1434  wral 2353  wrex 2354  {crab 2357  Vcvv 2612  cdif 2981  c0 3269  {csn 3422   class class class wbr 3811  ωcom 4368  cen 6385  Fincfn 6387
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-sbc 2827  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-br 3812  df-opab 3866  df-id 4084  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-1o 6113  df-er 6222  df-en 6388  df-fin 6390
This theorem is referenced by: (None)
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