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Theorem regexmidlem1 4202
 Description: Lemma for regexmid 4203. 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 2028 . . . . . . 7 (x = y → (x = {∅} ↔ y = {∅}))
2 eqeq1 2028 . . . . . . . 8 (x = y → (x = ∅ ↔ y = ∅))
32anbi1d 441 . . . . . . 7 (x = y → ((x = ∅ φ) ↔ (y = ∅ φ)))
41, 3orbi12d 694 . . . . . 6 (x = y → ((x = {∅} (x = ∅ φ)) ↔ (y = {∅} (y = ∅ φ))))
5 regexmidlemm.a . . . . . 6 A = {x {∅, {∅}} ∣ (x = {∅} (x = ∅ φ))}
64, 5elrab2 2677 . . . . 5 (y A ↔ (y {∅, {∅}} (y = {∅} (y = ∅ φ))))
76simprbi 260 . . . 4 (y A → (y = {∅} (y = ∅ φ)))
8 0ex 3858 . . . . . . . . 9 V
98snid 3377 . . . . . . . 8 {∅}
10 eleq2 2083 . . . . . . . 8 (y = {∅} → (∅ y ↔ ∅ {∅}))
119, 10mpbiri 157 . . . . . . 7 (y = {∅} → ∅ y)
12 eleq1 2082 . . . . . . . . 9 (z = ∅ → (z y ↔ ∅ y))
13 eleq1 2082 . . . . . . . . . 10 (z = ∅ → (z A ↔ ∅ A))
1413notbid 579 . . . . . . . . 9 (z = ∅ → (¬ z A ↔ ¬ ∅ A))
1512, 14imbi12d 223 . . . . . . . 8 (z = ∅ → ((z y → ¬ z A) ↔ (∅ y → ¬ ∅ A)))
168, 15spcv 2623 . . . . . . 7 (z(z y → ¬ z A) → (∅ y → ¬ ∅ A))
1711, 16syl5com 26 . . . . . 6 (y = {∅} → (z(z y → ¬ z A) → ¬ ∅ A))
188prid1 3450 . . . . . . . . . 10 {∅, {∅}}
19 eqeq1 2028 . . . . . . . . . . . 12 (x = ∅ → (x = {∅} ↔ ∅ = {∅}))
20 eqeq1 2028 . . . . . . . . . . . . 13 (x = ∅ → (x = ∅ ↔ ∅ = ∅))
2120anbi1d 441 . . . . . . . . . . . 12 (x = ∅ → ((x = ∅ φ) ↔ (∅ = ∅ φ)))
2219, 21orbi12d 694 . . . . . . . . . . 11 (x = ∅ → ((x = {∅} (x = ∅ φ)) ↔ (∅ = {∅} (∅ = ∅ φ))))
2322, 5elrab2 2677 . . . . . . . . . 10 (∅ A ↔ (∅ {∅, {∅}} (∅ = {∅} (∅ = ∅ φ))))
2418, 23mpbiran 835 . . . . . . . . 9 (∅ A ↔ (∅ = {∅} (∅ = ∅ φ)))
25 pm2.46 645 . . . . . . . . 9 (¬ (∅ = {∅} (∅ = ∅ φ)) → ¬ (∅ = ∅ φ))
2624, 25sylnbi 590 . . . . . . . 8 (¬ ∅ A → ¬ (∅ = ∅ φ))
27 eqid 2022 . . . . . . . . 9 ∅ = ∅
2827biantrur 287 . . . . . . . 8 (φ ↔ (∅ = ∅ φ))
2926, 28sylnibr 589 . . . . . . 7 (¬ ∅ A → ¬ φ)
3029olcd 640 . . . . . 6 (¬ ∅ A → (φ ¬ φ))
3117, 30syl6 29 . . . . 5 (y = {∅} → (z(z y → ¬ z A) → (φ ¬ φ)))
32 orc 620 . . . . . . 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 623 . . . 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 1471 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 616  ∀wal 1226   = wceq 1228  ∃wex 1362   ∈ wcel 1374  {crab 2288  ∅c0 3201  {csn 3350  {cpr 3351 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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-nul 3857 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-nul 3202  df-sn 3356  df-pr 3357 This theorem is referenced by:  regexmid  4203  nnregexmid  4269
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