ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  regexmidlem1 Structured version   GIF version

Theorem regexmidlem1 4218
Description: Lemma for regexmid 4219. If A has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2019.)
Hypothesis
Ref Expression
regexmidlemm.a A = {x {∅, {∅}} ∣ (x = {∅} (x = ∅ φ))}
Assertion
Ref Expression
regexmidlem1 (y(y A z(z y → ¬ z A)) → (φ ¬ φ))
Distinct variable groups:   y,A,z   φ,x,y
Allowed substitution hints:   φ(z)   A(x)

Proof of Theorem regexmidlem1
StepHypRef Expression
1 eqeq1 2043 . . . . . . 7 (x = y → (x = {∅} ↔ y = {∅}))
2 eqeq1 2043 . . . . . . . 8 (x = y → (x = ∅ ↔ y = ∅))
32anbi1d 438 . . . . . . 7 (x = y → ((x = ∅ φ) ↔ (y = ∅ φ)))
41, 3orbi12d 706 . . . . . 6 (x = y → ((x = {∅} (x = ∅ φ)) ↔ (y = {∅} (y = ∅ φ))))
5 regexmidlemm.a . . . . . 6 A = {x {∅, {∅}} ∣ (x = {∅} (x = ∅ φ))}
64, 5elrab2 2694 . . . . 5 (y A ↔ (y {∅, {∅}} (y = {∅} (y = ∅ φ))))
76simprbi 260 . . . 4 (y A → (y = {∅} (y = ∅ φ)))
8 0ex 3875 . . . . . . . . 9 V
98snid 3394 . . . . . . . 8 {∅}
10 eleq2 2098 . . . . . . . 8 (y = {∅} → (∅ y ↔ ∅ {∅}))
119, 10mpbiri 157 . . . . . . 7 (y = {∅} → ∅ y)
12 eleq1 2097 . . . . . . . . 9 (z = ∅ → (z y ↔ ∅ y))
13 eleq1 2097 . . . . . . . . . 10 (z = ∅ → (z A ↔ ∅ A))
1413notbid 591 . . . . . . . . 9 (z = ∅ → (¬ z A ↔ ¬ ∅ A))
1512, 14imbi12d 223 . . . . . . . 8 (z = ∅ → ((z y → ¬ z A) ↔ (∅ y → ¬ ∅ A)))
168, 15spcv 2640 . . . . . . 7 (z(z y → ¬ z A) → (∅ y → ¬ ∅ A))
1711, 16syl5com 26 . . . . . 6 (y = {∅} → (z(z y → ¬ z A) → ¬ ∅ A))
188prid1 3467 . . . . . . . . . 10 {∅, {∅}}
19 eqeq1 2043 . . . . . . . . . . . 12 (x = ∅ → (x = {∅} ↔ ∅ = {∅}))
20 eqeq1 2043 . . . . . . . . . . . . 13 (x = ∅ → (x = ∅ ↔ ∅ = ∅))
2120anbi1d 438 . . . . . . . . . . . 12 (x = ∅ → ((x = ∅ φ) ↔ (∅ = ∅ φ)))
2219, 21orbi12d 706 . . . . . . . . . . 11 (x = ∅ → ((x = {∅} (x = ∅ φ)) ↔ (∅ = {∅} (∅ = ∅ φ))))
2322, 5elrab2 2694 . . . . . . . . . 10 (∅ A ↔ (∅ {∅, {∅}} (∅ = {∅} (∅ = ∅ φ))))
2418, 23mpbiran 846 . . . . . . . . 9 (∅ A ↔ (∅ = {∅} (∅ = ∅ φ)))
25 pm2.46 657 . . . . . . . . 9 (¬ (∅ = {∅} (∅ = ∅ φ)) → ¬ (∅ = ∅ φ))
2624, 25sylnbi 602 . . . . . . . 8 (¬ ∅ A → ¬ (∅ = ∅ φ))
27 eqid 2037 . . . . . . . . 9 ∅ = ∅
2827biantrur 287 . . . . . . . 8 (φ ↔ (∅ = ∅ φ))
2926, 28sylnibr 601 . . . . . . 7 (¬ ∅ A → ¬ φ)
3029olcd 652 . . . . . 6 (¬ ∅ A → (φ ¬ φ))
3117, 30syl6 29 . . . . 5 (y = {∅} → (z(z y → ¬ z A) → (φ ¬ φ)))
32 orc 632 . . . . . . 7 (φ → (φ ¬ φ))
3332adantl 262 . . . . . 6 ((y = ∅ φ) → (φ ¬ φ))
3433a1d 22 . . . . 5 ((y = ∅ φ) → (z(z y → ¬ z A) → (φ ¬ φ)))
3531, 34jaoi 635 . . . 4 ((y = {∅} (y = ∅ φ)) → (z(z y → ¬ z A) → (φ ¬ φ)))
367, 35syl 14 . . 3 (y A → (z(z y → ¬ z A) → (φ ¬ φ)))
3736imp 115 . 2 ((y A z(z y → ¬ z A)) → (φ ¬ φ))
3837exlimiv 1486 1 (y(y A z(z y → ¬ z A)) → (φ ¬ φ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   wo 628  wal 1240   = wceq 1242  wex 1378   wcel 1390  {crab 2304  c0 3218  {csn 3367  {cpr 3368
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-nul 3219  df-sn 3373  df-pr 3374
This theorem is referenced by:  regexmid  4219  nnregexmid  4285
  Copyright terms: Public domain W3C validator