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Theorem regexmid 4203
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 4204. (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 2022 . . 3 {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))}
21regexmidlemm 4201 . 2 y y {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))}
3 pp0ex 3914 . . . 4 {∅, {∅}} V
43rabex 3875 . . 3 {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} V
5 eleq2 2083 . . . . 5 (x = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → (y xy {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))}))
65exbidv 1688 . . . 4 (x = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → (y y xy y {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))}))
7 eleq2 2083 . . . . . . . . 9 (x = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → (z xz {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))}))
87notbid 579 . . . . . . . 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 1687 . . . . . 6 (x = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → (z(z y → ¬ z x) ↔ z(z y → ¬ z {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))})))
115, 10anbi12d 445 . . . . 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 1688 . . . 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 2585 . 2 (y y {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → y(y {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} z(z y → ¬ z {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))})))
161regexmidlem1 4202 . 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 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-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-nul 3857  ax-pow 3901
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-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357
This theorem is referenced by: (None)
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