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Theorem exmidmotap 7321
Description: The proposition that every class has at most one tight apartness is equivalent to excluded middle. (Contributed by Jim Kingdon, 14-Feb-2025.)
Assertion
Ref Expression
exmidmotap (EXMID ↔ ∀𝑥∃*𝑟 𝑟 TAp 𝑥)
Distinct variable group:   𝑥,𝑟

Proof of Theorem exmidmotap
Dummy variables 𝑠 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 529 . . . . . . . 8 ((EXMID ∧ (𝑟 TAp 𝑥𝑠 TAp 𝑥)) → 𝑟 TAp 𝑥)
2 exmidapne 7320 . . . . . . . . 9 (EXMID → (𝑟 TAp 𝑥𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝑥𝑣𝑥) ∧ 𝑢𝑣)}))
32adantr 276 . . . . . . . 8 ((EXMID ∧ (𝑟 TAp 𝑥𝑠 TAp 𝑥)) → (𝑟 TAp 𝑥𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝑥𝑣𝑥) ∧ 𝑢𝑣)}))
41, 3mpbid 147 . . . . . . 7 ((EXMID ∧ (𝑟 TAp 𝑥𝑠 TAp 𝑥)) → 𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝑥𝑣𝑥) ∧ 𝑢𝑣)})
5 simprr 531 . . . . . . . 8 ((EXMID ∧ (𝑟 TAp 𝑥𝑠 TAp 𝑥)) → 𝑠 TAp 𝑥)
6 exmidapne 7320 . . . . . . . . 9 (EXMID → (𝑠 TAp 𝑥𝑠 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝑥𝑣𝑥) ∧ 𝑢𝑣)}))
76adantr 276 . . . . . . . 8 ((EXMID ∧ (𝑟 TAp 𝑥𝑠 TAp 𝑥)) → (𝑠 TAp 𝑥𝑠 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝑥𝑣𝑥) ∧ 𝑢𝑣)}))
85, 7mpbid 147 . . . . . . 7 ((EXMID ∧ (𝑟 TAp 𝑥𝑠 TAp 𝑥)) → 𝑠 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝑥𝑣𝑥) ∧ 𝑢𝑣)})
94, 8eqtr4d 2229 . . . . . 6 ((EXMID ∧ (𝑟 TAp 𝑥𝑠 TAp 𝑥)) → 𝑟 = 𝑠)
109ex 115 . . . . 5 (EXMID → ((𝑟 TAp 𝑥𝑠 TAp 𝑥) → 𝑟 = 𝑠))
1110alrimivv 1886 . . . 4 (EXMID → ∀𝑟𝑠((𝑟 TAp 𝑥𝑠 TAp 𝑥) → 𝑟 = 𝑠))
12 tapeq1 7312 . . . . 5 (𝑟 = 𝑠 → (𝑟 TAp 𝑥𝑠 TAp 𝑥))
1312mo4 2103 . . . 4 (∃*𝑟 𝑟 TAp 𝑥 ↔ ∀𝑟𝑠((𝑟 TAp 𝑥𝑠 TAp 𝑥) → 𝑟 = 𝑠))
1411, 13sylibr 134 . . 3 (EXMID → ∃*𝑟 𝑟 TAp 𝑥)
1514alrimiv 1885 . 2 (EXMID → ∀𝑥∃*𝑟 𝑟 TAp 𝑥)
16 2onn 6574 . . . 4 2o ∈ ω
17 tapeq2 7313 . . . . . 6 (𝑥 = 2o → (𝑟 TAp 𝑥𝑟 TAp 2o))
1817mobidv 2078 . . . . 5 (𝑥 = 2o → (∃*𝑟 𝑟 TAp 𝑥 ↔ ∃*𝑟 𝑟 TAp 2o))
1918spcgv 2847 . . . 4 (2o ∈ ω → (∀𝑥∃*𝑟 𝑟 TAp 𝑥 → ∃*𝑟 𝑟 TAp 2o))
2016, 19ax-mp 5 . . 3 (∀𝑥∃*𝑟 𝑟 TAp 𝑥 → ∃*𝑟 𝑟 TAp 2o)
21 2omotap 7319 . . 3 (∃*𝑟 𝑟 TAp 2oEXMID)
2220, 21syl 14 . 2 (∀𝑥∃*𝑟 𝑟 TAp 𝑥EXMID)
2315, 22impbii 126 1 (EXMID ↔ ∀𝑥∃*𝑟 𝑟 TAp 𝑥)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1362   = wceq 1364  ∃*wmo 2043  wcel 2164  wne 2364  {copab 4089  EXMIDwem 4223  ωcom 4622  2oc2o 6463   TAp wtap 7309
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-exmid 4224  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fo 5260  df-fv 5262  df-1st 6193  df-2nd 6194  df-1o 6469  df-2o 6470  df-pap 7308  df-tap 7310
This theorem is referenced by: (None)
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