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Theorem diffitest 6375
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 3912 . . . . . 6 ∅ ∈ V
2 snfig 6322 . . . . . 6 (∅ ∈ V → {∅} ∈ Fin)
31, 2ax-mp 7 . . . . 5 {∅} ∈ Fin
4 diffitest.1 . . . . 5 𝑎 ∈ Fin ∀𝑏(𝑎𝑏) ∈ Fin
5 difeq1 3083 . . . . . . . 8 (𝑎 = {∅} → (𝑎𝑏) = ({∅} ∖ 𝑏))
65eleq1d 2122 . . . . . . 7 (𝑎 = {∅} → ((𝑎𝑏) ∈ Fin ↔ ({∅} ∖ 𝑏) ∈ Fin))
76albidv 1721 . . . . . 6 (𝑎 = {∅} → (∀𝑏(𝑎𝑏) ∈ Fin ↔ ∀𝑏({∅} ∖ 𝑏) ∈ Fin))
87rspcv 2669 . . . . 5 ({∅} ∈ Fin → (∀𝑎 ∈ Fin ∀𝑏(𝑎𝑏) ∈ Fin → ∀𝑏({∅} ∖ 𝑏) ∈ Fin))
93, 4, 8mp2 16 . . . 4 𝑏({∅} ∖ 𝑏) ∈ Fin
10 rabexg 3928 . . . . . 6 ({∅} ∈ Fin → {𝑥 ∈ {∅} ∣ 𝜑} ∈ V)
113, 10ax-mp 7 . . . . 5 {𝑥 ∈ {∅} ∣ 𝜑} ∈ V
12 difeq2 3084 . . . . . 6 (𝑏 = {𝑥 ∈ {∅} ∣ 𝜑} → ({∅} ∖ 𝑏) = ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
1312eleq1d 2122 . . . . 5 (𝑏 = {𝑥 ∈ {∅} ∣ 𝜑} → (({∅} ∖ 𝑏) ∈ Fin ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin))
1411, 13spcv 2663 . . . 4 (∀𝑏({∅} ∖ 𝑏) ∈ Fin → ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin)
159, 14ax-mp 7 . . 3 ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin
16 isfi 6272 . . 3 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin ↔ ∃𝑛 ∈ ω ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛)
1715, 16mpbi 137 . 2 𝑛 ∈ ω ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛
18 0elnn 4368 . . . . 5 (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
19 breq2 3796 . . . . . . . . . 10 (𝑛 = ∅ → (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ ∅))
20 en0 6306 . . . . . . . . . 10 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ ∅ ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅)
2119, 20syl6bb 189 . . . . . . . . 9 (𝑛 = ∅ → (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅))
2221biimpac 286 . . . . . . . 8 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 = ∅) → ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅)
23 rabeq0 3275 . . . . . . . . 9 ({𝑥 ∈ {∅} ∣ ¬ 𝜑} = ∅ ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑)
24 notrab 3242 . . . . . . . . . 10 ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = {𝑥 ∈ {∅} ∣ ¬ 𝜑}
2524eqeq1i 2063 . . . . . . . . 9 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅ ↔ {𝑥 ∈ {∅} ∣ ¬ 𝜑} = ∅)
261snm 3516 . . . . . . . . . 10 𝑤 𝑤 ∈ {∅}
27 r19.3rmv 3340 . . . . . . . . . 10 (∃𝑤 𝑤 ∈ {∅} → (¬ ¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑))
2826, 27ax-mp 7 . . . . . . . . 9 (¬ ¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑)
2923, 25, 283bitr4i 205 . . . . . . . 8 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅ ↔ ¬ ¬ 𝜑)
3022, 29sylib 131 . . . . . . 7 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 = ∅) → ¬ ¬ 𝜑)
3130olcd 663 . . . . . 6 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 = ∅) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
32 ensym 6292 . . . . . . . 8 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 ≈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
33 elex2 2587 . . . . . . . 8 (∅ ∈ 𝑛 → ∃𝑤 𝑤𝑛)
34 enm 6325 . . . . . . . 8 ((𝑛 ≈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∧ ∃𝑤 𝑤𝑛) → ∃𝑦 𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
3532, 33, 34syl2an 277 . . . . . . 7 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ ∅ ∈ 𝑛) → ∃𝑦 𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
36 biidd 165 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜑))
3736elrab 2721 . . . . . . . . . . 11 (𝑦 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ↔ (𝑦 ∈ {∅} ∧ ¬ 𝜑))
3837simprbi 264 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → ¬ 𝜑)
3938orcd 662 . . . . . . . . 9 (𝑦 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4039, 24eleq2s 2148 . . . . . . . 8 (𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4140exlimiv 1505 . . . . . . 7 (∃𝑦 𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4235, 41syl 14 . . . . . 6 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ ∅ ∈ 𝑛) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4331, 42jaodan 721 . . . . 5 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ (𝑛 = ∅ ∨ ∅ ∈ 𝑛)) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4418, 43sylan2 274 . . . 4 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 ∈ ω) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4544ancoms 259 . . 3 ((𝑛 ∈ ω ∧ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4645rexlimiva 2445 . 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 101  wb 102  wo 639  wal 1257   = wceq 1259  wex 1397  wcel 1409  wral 2323  wrex 2324  {crab 2327  Vcvv 2574  cdif 2942  c0 3252  {csn 3403   class class class wbr 3792  ωcom 4341  cen 6250  Fincfn 6252
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 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  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-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-br 3793  df-opab 3847  df-id 4058  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-1o 6032  df-er 6137  df-en 6253  df-fin 6255
This theorem is referenced by: (None)
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