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Theorem diffitest 6900
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 4142 . . . . . 6 ∅ ∈ V
2 snfig 6827 . . . . . 6 (∅ ∈ V → {∅} ∈ Fin)
31, 2ax-mp 5 . . . . 5 {∅} ∈ Fin
4 diffitest.1 . . . . 5 𝑎 ∈ Fin ∀𝑏(𝑎𝑏) ∈ Fin
5 difeq1 3258 . . . . . . . 8 (𝑎 = {∅} → (𝑎𝑏) = ({∅} ∖ 𝑏))
65eleq1d 2256 . . . . . . 7 (𝑎 = {∅} → ((𝑎𝑏) ∈ Fin ↔ ({∅} ∖ 𝑏) ∈ Fin))
76albidv 1834 . . . . . 6 (𝑎 = {∅} → (∀𝑏(𝑎𝑏) ∈ Fin ↔ ∀𝑏({∅} ∖ 𝑏) ∈ Fin))
87rspcv 2849 . . . . 5 ({∅} ∈ Fin → (∀𝑎 ∈ Fin ∀𝑏(𝑎𝑏) ∈ Fin → ∀𝑏({∅} ∖ 𝑏) ∈ Fin))
93, 4, 8mp2 16 . . . 4 𝑏({∅} ∖ 𝑏) ∈ Fin
10 rabexg 4158 . . . . . 6 ({∅} ∈ Fin → {𝑥 ∈ {∅} ∣ 𝜑} ∈ V)
113, 10ax-mp 5 . . . . 5 {𝑥 ∈ {∅} ∣ 𝜑} ∈ V
12 difeq2 3259 . . . . . 6 (𝑏 = {𝑥 ∈ {∅} ∣ 𝜑} → ({∅} ∖ 𝑏) = ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
1312eleq1d 2256 . . . . 5 (𝑏 = {𝑥 ∈ {∅} ∣ 𝜑} → (({∅} ∖ 𝑏) ∈ Fin ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin))
1411, 13spcv 2843 . . . 4 (∀𝑏({∅} ∖ 𝑏) ∈ Fin → ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin)
159, 14ax-mp 5 . . 3 ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin
16 isfi 6774 . . 3 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin ↔ ∃𝑛 ∈ ω ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛)
1715, 16mpbi 145 . 2 𝑛 ∈ ω ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛
18 0elnn 4630 . . . . 5 (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
19 breq2 4019 . . . . . . . . . 10 (𝑛 = ∅ → (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ ∅))
20 en0 6808 . . . . . . . . . 10 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ ∅ ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅)
2119, 20bitrdi 196 . . . . . . . . 9 (𝑛 = ∅ → (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅))
2221biimpac 298 . . . . . . . 8 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 = ∅) → ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅)
23 rabeq0 3464 . . . . . . . . 9 ({𝑥 ∈ {∅} ∣ ¬ 𝜑} = ∅ ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑)
24 notrab 3424 . . . . . . . . . 10 ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = {𝑥 ∈ {∅} ∣ ¬ 𝜑}
2524eqeq1i 2195 . . . . . . . . 9 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅ ↔ {𝑥 ∈ {∅} ∣ ¬ 𝜑} = ∅)
261snm 3724 . . . . . . . . . 10 𝑤 𝑤 ∈ {∅}
27 r19.3rmv 3525 . . . . . . . . . 10 (∃𝑤 𝑤 ∈ {∅} → (¬ ¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑))
2826, 27ax-mp 5 . . . . . . . . 9 (¬ ¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑)
2923, 25, 283bitr4i 212 . . . . . . . 8 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅ ↔ ¬ ¬ 𝜑)
3022, 29sylib 122 . . . . . . 7 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 = ∅) → ¬ ¬ 𝜑)
3130olcd 735 . . . . . 6 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 = ∅) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
32 ensym 6794 . . . . . . . 8 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 ≈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
33 elex2 2765 . . . . . . . 8 (∅ ∈ 𝑛 → ∃𝑤 𝑤𝑛)
34 enm 6833 . . . . . . . 8 ((𝑛 ≈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∧ ∃𝑤 𝑤𝑛) → ∃𝑦 𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
3532, 33, 34syl2an 289 . . . . . . 7 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ ∅ ∈ 𝑛) → ∃𝑦 𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
36 biidd 172 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜑))
3736elrab 2905 . . . . . . . . . . 11 (𝑦 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ↔ (𝑦 ∈ {∅} ∧ ¬ 𝜑))
3837simprbi 275 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → ¬ 𝜑)
3938orcd 734 . . . . . . . . 9 (𝑦 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4039, 24eleq2s 2282 . . . . . . . 8 (𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4140exlimiv 1608 . . . . . . 7 (∃𝑦 𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4235, 41syl 14 . . . . . 6 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ ∅ ∈ 𝑛) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4331, 42jaodan 798 . . . . 5 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ (𝑛 = ∅ ∨ ∅ ∈ 𝑛)) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4418, 43sylan2 286 . . . 4 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 ∈ ω) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4544ancoms 268 . . 3 ((𝑛 ∈ ω ∧ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4645rexlimiva 2599 . 2 (∃𝑛 ∈ ω ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4717, 46ax-mp 5 1 𝜑 ∨ ¬ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104  wb 105  wo 709  wal 1361   = wceq 1363  wex 1502  wcel 2158  wral 2465  wrex 2466  {crab 2469  Vcvv 2749  cdif 3138  c0 3434  {csn 3604   class class class wbr 4015  ωcom 4601  cen 6751  Fincfn 6753
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-id 4305  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-1o 6430  df-er 6548  df-en 6754  df-fin 6756
This theorem is referenced by: (None)
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