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Theorem regexmid 4218
Description: The axiom of foundation implies excluded middle.

By foundation (or regularity), we mean the principle that every inhabited set has an element which is minimal (when arranged by ). The statement of foundation here is taken from Metamath Proof Explorer's ax-reg, and is identical (modulo one unnecessary quantifier) to the statement of foundation in Theorem "Foundation implies instances of EM" of [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic".

For this reason, IZF does not adopt foundation as an axiom and instead replaces it with ax-setind 4219. (Contributed by Jim Kingdon, 3-Sep-2019.)

Hypothesis
Ref Expression
regexmid.1 (y y xy(y x z(z y → ¬ z x)))
Assertion
Ref Expression
regexmid (φ ¬ φ)
Distinct variable group:   φ,x,y,z

Proof of Theorem regexmid
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 eqid 2037 . . 3 {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))}
21regexmidlemm 4216 . 2 y y {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))}
3 pp0ex 3930 . . . 4 {∅, {∅}} V
43rabex 3891 . . 3 {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} V
5 eleq2 2098 . . . . 5 (x = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → (y xy {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))}))
65exbidv 1703 . . . 4 (x = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → (y y xy y {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))}))
7 eleq2 2098 . . . . . . . . 9 (x = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → (z xz {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))}))
87notbid 591 . . . . . . . 8 (x = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → (¬ z x ↔ ¬ z {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))}))
98imbi2d 219 . . . . . . 7 (x = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → ((z y → ¬ z x) ↔ (z y → ¬ z {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))})))
109albidv 1702 . . . . . 6 (x = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → (z(z y → ¬ z x) ↔ z(z y → ¬ z {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))})))
115, 10anbi12d 442 . . . . 5 (x = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → ((y x z(z y → ¬ z x)) ↔ (y {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} z(z y → ¬ z {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))}))))
1211exbidv 1703 . . . 4 (x = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → (y(y x z(z y → ¬ z x)) ↔ y(y {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} z(z y → ¬ z {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))}))))
136, 12imbi12d 223 . . 3 (x = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → ((y y xy(y x z(z y → ¬ z x))) ↔ (y y {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → y(y {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} z(z y → ¬ z {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))})))))
14 regexmid.1 . . 3 (y y xy(y x z(z y → ¬ z x)))
154, 13, 14vtocl 2602 . 2 (y y {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → y(y {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} z(z y → ¬ z {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))})))
161regexmidlem1 4217 . 2 (y(y {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} z(z y → ¬ z {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))})) → (φ ¬ φ))
172, 15, 16mp2b 8 1 (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 3366  {cpr 3367
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-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-nul 3873  ax-pow 3917
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-in 2918  df-ss 2925  df-nul 3219  df-pw 3352  df-sn 3372  df-pr 3373
This theorem is referenced by: (None)
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