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Theorem diffitest 6657
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 3972 . . . . . 6 ∅ ∈ V
2 snfig 6585 . . . . . 6 (∅ ∈ V → {∅} ∈ Fin)
31, 2ax-mp 7 . . . . 5 {∅} ∈ Fin
4 diffitest.1 . . . . 5 𝑎 ∈ Fin ∀𝑏(𝑎𝑏) ∈ Fin
5 difeq1 3112 . . . . . . . 8 (𝑎 = {∅} → (𝑎𝑏) = ({∅} ∖ 𝑏))
65eleq1d 2157 . . . . . . 7 (𝑎 = {∅} → ((𝑎𝑏) ∈ Fin ↔ ({∅} ∖ 𝑏) ∈ Fin))
76albidv 1753 . . . . . 6 (𝑎 = {∅} → (∀𝑏(𝑎𝑏) ∈ Fin ↔ ∀𝑏({∅} ∖ 𝑏) ∈ Fin))
87rspcv 2719 . . . . 5 ({∅} ∈ Fin → (∀𝑎 ∈ Fin ∀𝑏(𝑎𝑏) ∈ Fin → ∀𝑏({∅} ∖ 𝑏) ∈ Fin))
93, 4, 8mp2 16 . . . 4 𝑏({∅} ∖ 𝑏) ∈ Fin
10 rabexg 3988 . . . . . 6 ({∅} ∈ Fin → {𝑥 ∈ {∅} ∣ 𝜑} ∈ V)
113, 10ax-mp 7 . . . . 5 {𝑥 ∈ {∅} ∣ 𝜑} ∈ V
12 difeq2 3113 . . . . . 6 (𝑏 = {𝑥 ∈ {∅} ∣ 𝜑} → ({∅} ∖ 𝑏) = ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
1312eleq1d 2157 . . . . 5 (𝑏 = {𝑥 ∈ {∅} ∣ 𝜑} → (({∅} ∖ 𝑏) ∈ Fin ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin))
1411, 13spcv 2713 . . . 4 (∀𝑏({∅} ∖ 𝑏) ∈ Fin → ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin)
159, 14ax-mp 7 . . 3 ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin
16 isfi 6532 . . 3 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin ↔ ∃𝑛 ∈ ω ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛)
1715, 16mpbi 144 . 2 𝑛 ∈ ω ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛
18 0elnn 4445 . . . . 5 (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
19 breq2 3855 . . . . . . . . . 10 (𝑛 = ∅ → (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ ∅))
20 en0 6566 . . . . . . . . . 10 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ ∅ ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅)
2119, 20syl6bb 195 . . . . . . . . 9 (𝑛 = ∅ → (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅))
2221biimpac 293 . . . . . . . 8 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 = ∅) → ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅)
23 rabeq0 3316 . . . . . . . . 9 ({𝑥 ∈ {∅} ∣ ¬ 𝜑} = ∅ ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑)
24 notrab 3277 . . . . . . . . . 10 ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = {𝑥 ∈ {∅} ∣ ¬ 𝜑}
2524eqeq1i 2096 . . . . . . . . 9 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅ ↔ {𝑥 ∈ {∅} ∣ ¬ 𝜑} = ∅)
261snm 3566 . . . . . . . . . 10 𝑤 𝑤 ∈ {∅}
27 r19.3rmv 3376 . . . . . . . . . 10 (∃𝑤 𝑤 ∈ {∅} → (¬ ¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑))
2826, 27ax-mp 7 . . . . . . . . 9 (¬ ¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑)
2923, 25, 283bitr4i 211 . . . . . . . 8 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅ ↔ ¬ ¬ 𝜑)
3022, 29sylib 121 . . . . . . 7 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 = ∅) → ¬ ¬ 𝜑)
3130olcd 689 . . . . . 6 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 = ∅) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
32 ensym 6552 . . . . . . . 8 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 ≈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
33 elex2 2636 . . . . . . . 8 (∅ ∈ 𝑛 → ∃𝑤 𝑤𝑛)
34 enm 6590 . . . . . . . 8 ((𝑛 ≈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∧ ∃𝑤 𝑤𝑛) → ∃𝑦 𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
3532, 33, 34syl2an 284 . . . . . . 7 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ ∅ ∈ 𝑛) → ∃𝑦 𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
36 biidd 171 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜑))
3736elrab 2772 . . . . . . . . . . 11 (𝑦 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ↔ (𝑦 ∈ {∅} ∧ ¬ 𝜑))
3837simprbi 270 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → ¬ 𝜑)
3938orcd 688 . . . . . . . . 9 (𝑦 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4039, 24eleq2s 2183 . . . . . . . 8 (𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4140exlimiv 1535 . . . . . . 7 (∃𝑦 𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4235, 41syl 14 . . . . . 6 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ ∅ ∈ 𝑛) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4331, 42jaodan 747 . . . . 5 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ (𝑛 = ∅ ∨ ∅ ∈ 𝑛)) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4418, 43sylan2 281 . . . 4 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 ∈ ω) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4544ancoms 265 . . 3 ((𝑛 ∈ ω ∧ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4645rexlimiva 2485 . 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 103  wb 104  wo 665  wal 1288   = wceq 1290  wex 1427  wcel 1439  wral 2360  wrex 2361  {crab 2364  Vcvv 2620  cdif 2997  c0 3287  {csn 3450   class class class wbr 3851  ωcom 4418  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-iinf 4416
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  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-ral 2365  df-rex 2366  df-rab 2369  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-id 4129  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-res 4464  df-ima 4465  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-1o 6195  df-er 6306  df-en 6512  df-fin 6514
This theorem is referenced by: (None)
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