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Theorem diffitest 6344
 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 → (𝐴 ∖ 𝐵) ∈ 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 3884 . . . . . 6 ∅ ∈ V
2 snfig 6291 . . . . . 6 (∅ ∈ V → {∅} ∈ Fin)
31, 2ax-mp 7 . . . . 5 {∅} ∈ Fin
4 diffitest.1 . . . . 5 𝑎 ∈ Fin ∀𝑏(𝑎𝑏) ∈ Fin
5 difeq1 3055 . . . . . . . 8 (𝑎 = {∅} → (𝑎𝑏) = ({∅} ∖ 𝑏))
65eleq1d 2106 . . . . . . 7 (𝑎 = {∅} → ((𝑎𝑏) ∈ Fin ↔ ({∅} ∖ 𝑏) ∈ Fin))
76albidv 1705 . . . . . 6 (𝑎 = {∅} → (∀𝑏(𝑎𝑏) ∈ Fin ↔ ∀𝑏({∅} ∖ 𝑏) ∈ Fin))
87rspcv 2652 . . . . 5 ({∅} ∈ Fin → (∀𝑎 ∈ Fin ∀𝑏(𝑎𝑏) ∈ Fin → ∀𝑏({∅} ∖ 𝑏) ∈ Fin))
93, 4, 8mp2 16 . . . 4 𝑏({∅} ∖ 𝑏) ∈ Fin
10 rabexg 3900 . . . . . 6 ({∅} ∈ Fin → {𝑥 ∈ {∅} ∣ 𝜑} ∈ V)
113, 10ax-mp 7 . . . . 5 {𝑥 ∈ {∅} ∣ 𝜑} ∈ V
12 difeq2 3056 . . . . . 6 (𝑏 = {𝑥 ∈ {∅} ∣ 𝜑} → ({∅} ∖ 𝑏) = ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
1312eleq1d 2106 . . . . 5 (𝑏 = {𝑥 ∈ {∅} ∣ 𝜑} → (({∅} ∖ 𝑏) ∈ Fin ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin))
1411, 13spcv 2646 . . . 4 (∀𝑏({∅} ∖ 𝑏) ∈ Fin → ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin)
159, 14ax-mp 7 . . 3 ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin
16 isfi 6241 . . 3 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∈ Fin ↔ ∃𝑛 ∈ ω ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛)
1715, 16mpbi 133 . 2 𝑛 ∈ ω ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛
18 0elnn 4340 . . . . 5 (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
19 breq2 3768 . . . . . . . . . 10 (𝑛 = ∅ → (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ ∅))
20 en0 6275 . . . . . . . . . 10 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ ∅ ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅)
2119, 20syl6bb 185 . . . . . . . . 9 (𝑛 = ∅ → (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ↔ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅))
2221biimpac 282 . . . . . . . 8 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 = ∅) → ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅)
23 rabeq0 3247 . . . . . . . . 9 ({𝑥 ∈ {∅} ∣ ¬ 𝜑} = ∅ ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑)
24 notrab 3214 . . . . . . . . . 10 ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = {𝑥 ∈ {∅} ∣ ¬ 𝜑}
2524eqeq1i 2047 . . . . . . . . 9 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅ ↔ {𝑥 ∈ {∅} ∣ ¬ 𝜑} = ∅)
261snm 3488 . . . . . . . . . 10 𝑤 𝑤 ∈ {∅}
27 r19.3rmv 3312 . . . . . . . . . 10 (∃𝑤 𝑤 ∈ {∅} → (¬ ¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑))
2826, 27ax-mp 7 . . . . . . . . 9 (¬ ¬ 𝜑 ↔ ∀𝑥 ∈ {∅} ¬ ¬ 𝜑)
2923, 25, 283bitr4i 201 . . . . . . . 8 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) = ∅ ↔ ¬ ¬ 𝜑)
3022, 29sylib 127 . . . . . . 7 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 = ∅) → ¬ ¬ 𝜑)
3130olcd 653 . . . . . 6 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 = ∅) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
32 ensym 6261 . . . . . . . 8 (({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 ≈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
33 elex2 2570 . . . . . . . 8 (∅ ∈ 𝑛 → ∃𝑤 𝑤𝑛)
34 enm 6294 . . . . . . . 8 ((𝑛 ≈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ∧ ∃𝑤 𝑤𝑛) → ∃𝑦 𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
3532, 33, 34syl2an 273 . . . . . . 7 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ ∅ ∈ 𝑛) → ∃𝑦 𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}))
36 biidd 161 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜑))
3736elrab 2698 . . . . . . . . . . 11 (𝑦 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} ↔ (𝑦 ∈ {∅} ∧ ¬ 𝜑))
3837simprbi 260 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → ¬ 𝜑)
3938orcd 652 . . . . . . . . 9 (𝑦 ∈ {𝑥 ∈ {∅} ∣ ¬ 𝜑} → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4039, 24eleq2s 2132 . . . . . . . 8 (𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4140exlimiv 1489 . . . . . . 7 (∃𝑦 𝑦 ∈ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4235, 41syl 14 . . . . . 6 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ ∅ ∈ 𝑛) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4331, 42jaodan 710 . . . . 5 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛 ∧ (𝑛 = ∅ ∨ ∅ ∈ 𝑛)) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4418, 43sylan2 270 . . . 4 ((({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛𝑛 ∈ ω) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4544ancoms 255 . . 3 ((𝑛 ∈ ω ∧ ({∅} ∖ {𝑥 ∈ {∅} ∣ 𝜑}) ≈ 𝑛) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4645rexlimiva 2428 . 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 97   ↔ wb 98   ∨ wo 629  ∀wal 1241   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307  {crab 2310  Vcvv 2557   ∖ cdif 2914  ∅c0 3224  {csn 3375   class class class wbr 3764  ωcom 4313   ≈ cen 6219  Fincfn 6221 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-opab 3819  df-id 4030  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-1o 6001  df-er 6106  df-en 6222  df-fin 6224 This theorem is referenced by: (None)
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