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Theorem ordsoexmid 4224
Description: Weak linearity of ordinals implies the law of the excluded middle (that is, decidability of an arbitrary proposition). (Contributed by Mario Carneiro and Jim Kingdon, 29-Jan-2019.)
Hypothesis
Ref Expression
ordsoexmid.1 E Or On
Assertion
Ref Expression
ordsoexmid (φ ¬ φ)

Proof of Theorem ordsoexmid
Dummy variables x y z w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordtriexmidlem 4192 . . . . 5 {w {∅} ∣ φ} On
21elexi 2544 . . . 4 {w {∅} ∣ φ} V
32sucid 4103 . . 3 {w {∅} ∣ φ} suc {w {∅} ∣ φ}
41onsuci 4191 . . . 4 suc {w {∅} ∣ φ} On
5 suc0 4097 . . . . 5 suc ∅ = {∅}
6 0elon 4078 . . . . . 6 On
76onsuci 4191 . . . . 5 suc ∅ On
85, 7eqeltrri 2093 . . . 4 {∅} On
9 eleq1 2082 . . . . . . 7 (x = {w {∅} ∣ φ} → (x On ↔ {w {∅} ∣ φ} On))
1093anbi1d 1196 . . . . . 6 (x = {w {∅} ∣ φ} → ((x On suc {w {∅} ∣ φ} On {∅} On) ↔ ({w {∅} ∣ φ} On suc {w {∅} ∣ φ} On {∅} On)))
11 eleq1 2082 . . . . . . 7 (x = {w {∅} ∣ φ} → (x suc {w {∅} ∣ φ} ↔ {w {∅} ∣ φ} suc {w {∅} ∣ φ}))
12 eleq1 2082 . . . . . . . 8 (x = {w {∅} ∣ φ} → (x {∅} ↔ {w {∅} ∣ φ} {∅}))
1312orbi1d 692 . . . . . . 7 (x = {w {∅} ∣ φ} → ((x {∅} {∅} suc {w {∅} ∣ φ}) ↔ ({w {∅} ∣ φ} {∅} {∅} suc {w {∅} ∣ φ})))
1411, 13imbi12d 223 . . . . . 6 (x = {w {∅} ∣ φ} → ((x suc {w {∅} ∣ φ} → (x {∅} {∅} suc {w {∅} ∣ φ})) ↔ ({w {∅} ∣ φ} suc {w {∅} ∣ φ} → ({w {∅} ∣ φ} {∅} {∅} suc {w {∅} ∣ φ}))))
1510, 14imbi12d 223 . . . . 5 (x = {w {∅} ∣ φ} → (((x On suc {w {∅} ∣ φ} On {∅} On) → (x suc {w {∅} ∣ φ} → (x {∅} {∅} suc {w {∅} ∣ φ}))) ↔ (({w {∅} ∣ φ} On suc {w {∅} ∣ φ} On {∅} On) → ({w {∅} ∣ φ} suc {w {∅} ∣ φ} → ({w {∅} ∣ φ} {∅} {∅} suc {w {∅} ∣ φ})))))
164elexi 2544 . . . . . 6 suc {w {∅} ∣ φ} V
17 eleq1 2082 . . . . . . . 8 (y = suc {w {∅} ∣ φ} → (y On ↔ suc {w {∅} ∣ φ} On))
18173anbi2d 1197 . . . . . . 7 (y = suc {w {∅} ∣ φ} → ((x On y On {∅} On) ↔ (x On suc {w {∅} ∣ φ} On {∅} On)))
19 eleq2 2083 . . . . . . . 8 (y = suc {w {∅} ∣ φ} → (x yx suc {w {∅} ∣ φ}))
20 eleq2 2083 . . . . . . . . 9 (y = suc {w {∅} ∣ φ} → ({∅} y ↔ {∅} suc {w {∅} ∣ φ}))
2120orbi2d 691 . . . . . . . 8 (y = suc {w {∅} ∣ φ} → ((x {∅} {∅} y) ↔ (x {∅} {∅} suc {w {∅} ∣ φ})))
2219, 21imbi12d 223 . . . . . . 7 (y = suc {w {∅} ∣ φ} → ((x y → (x {∅} {∅} y)) ↔ (x suc {w {∅} ∣ φ} → (x {∅} {∅} suc {w {∅} ∣ φ}))))
2318, 22imbi12d 223 . . . . . 6 (y = suc {w {∅} ∣ φ} → (((x On y On {∅} On) → (x y → (x {∅} {∅} y))) ↔ ((x On suc {w {∅} ∣ φ} On {∅} On) → (x suc {w {∅} ∣ φ} → (x {∅} {∅} suc {w {∅} ∣ φ})))))
24 p0ex 3913 . . . . . . 7 {∅} V
25 eleq1 2082 . . . . . . . . 9 (z = {∅} → (z On ↔ {∅} On))
26253anbi3d 1198 . . . . . . . 8 (z = {∅} → ((x On y On z On) ↔ (x On y On {∅} On)))
27 eleq2 2083 . . . . . . . . . 10 (z = {∅} → (x zx {∅}))
28 eleq1 2082 . . . . . . . . . 10 (z = {∅} → (z y ↔ {∅} y))
2927, 28orbi12d 694 . . . . . . . . 9 (z = {∅} → ((x z z y) ↔ (x {∅} {∅} y)))
3029imbi2d 219 . . . . . . . 8 (z = {∅} → ((x y → (x z z y)) ↔ (x y → (x {∅} {∅} y))))
3126, 30imbi12d 223 . . . . . . 7 (z = {∅} → (((x On y On z On) → (x y → (x z z y))) ↔ ((x On y On {∅} On) → (x y → (x {∅} {∅} y)))))
32 ordsoexmid.1 . . . . . . . . . . 11 E Or On
33 df-iso 4008 . . . . . . . . . . 11 ( E Or On ↔ ( E Po On x On y On z On (x E y → (x E z z E y))))
3432, 33mpbi 133 . . . . . . . . . 10 ( E Po On x On y On z On (x E y → (x E z z E y)))
3534simpri 106 . . . . . . . . 9 x On y On z On (x E y → (x E z z E y))
36 epel 4003 . . . . . . . . . . . 12 (x E yx y)
37 epel 4003 . . . . . . . . . . . . 13 (x E zx z)
38 epel 4003 . . . . . . . . . . . . 13 (z E yz y)
3937, 38orbi12i 668 . . . . . . . . . . . 12 ((x E z z E y) ↔ (x z z y))
4036, 39imbi12i 228 . . . . . . . . . . 11 ((x E y → (x E z z E y)) ↔ (x y → (x z z y)))
41402ralbii 2310 . . . . . . . . . 10 (y On z On (x E y → (x E z z E y)) ↔ y On z On (x y → (x z z y)))
4241ralbii 2308 . . . . . . . . 9 (x On y On z On (x E y → (x E z z E y)) ↔ x On y On z On (x y → (x z z y)))
4335, 42mpbi 133 . . . . . . . 8 x On y On z On (x y → (x z z y))
4443rspec3 2387 . . . . . . 7 ((x On y On z On) → (x y → (x z z y)))
4524, 31, 44vtocl 2585 . . . . . 6 ((x On y On {∅} On) → (x y → (x {∅} {∅} y)))
4616, 23, 45vtocl 2585 . . . . 5 ((x On suc {w {∅} ∣ φ} On {∅} On) → (x suc {w {∅} ∣ φ} → (x {∅} {∅} suc {w {∅} ∣ φ})))
472, 15, 46vtocl 2585 . . . 4 (({w {∅} ∣ φ} On suc {w {∅} ∣ φ} On {∅} On) → ({w {∅} ∣ φ} suc {w {∅} ∣ φ} → ({w {∅} ∣ φ} {∅} {∅} suc {w {∅} ∣ φ})))
481, 4, 8, 47mp3an 1217 . . 3 ({w {∅} ∣ φ} suc {w {∅} ∣ φ} → ({w {∅} ∣ φ} {∅} {∅} suc {w {∅} ∣ φ}))
492elsnc 3373 . . . . 5 ({w {∅} ∣ φ} {∅} ↔ {w {∅} ∣ φ} = ∅)
50 ordtriexmidlem2 4193 . . . . 5 ({w {∅} ∣ φ} = ∅ → ¬ φ)
5149, 50sylbi 114 . . . 4 ({w {∅} ∣ φ} {∅} → ¬ φ)
52 elirr 4208 . . . . . . 7 ¬ {∅} {∅}
53 elrabi 2672 . . . . . . 7 ({∅} {w {∅} ∣ φ} → {∅} {∅})
5452, 53mto 575 . . . . . 6 ¬ {∅} {w {∅} ∣ φ}
55 elsuci 4089 . . . . . . 7 ({∅} suc {w {∅} ∣ φ} → ({∅} {w {∅} ∣ φ} {∅} = {w {∅} ∣ φ}))
5655ord 630 . . . . . 6 ({∅} suc {w {∅} ∣ φ} → (¬ {∅} {w {∅} ∣ φ} → {∅} = {w {∅} ∣ φ}))
5754, 56mpi 15 . . . . 5 ({∅} suc {w {∅} ∣ φ} → {∅} = {w {∅} ∣ φ})
58 0ex 3858 . . . . . . 7 V
59 biidd 161 . . . . . . 7 (w = ∅ → (φφ))
6058, 59rabsnt 3419 . . . . . 6 ({w {∅} ∣ φ} = {∅} → φ)
6160eqcoms 2025 . . . . 5 ({∅} = {w {∅} ∣ φ} → φ)
6257, 61syl 14 . . . 4 ({∅} suc {w {∅} ∣ φ} → φ)
6351, 62orim12i 663 . . 3 (({w {∅} ∣ φ} {∅} {∅} suc {w {∅} ∣ φ}) → (¬ φ φ))
643, 48, 63mp2b 8 . 2 φ φ)
65 orcom 634 . 2 ((¬ φ φ) ↔ (φ ¬ φ))
6664, 65mpbi 133 1 (φ ¬ φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   wo 616   w3a 873   = wceq 1228   wcel 1374  wral 2284  {crab 2288  c0 3201  {csn 3350   class class class wbr 3738   E cep 3998   Po wpo 4005   Or wor 4006  Oncon0 4049  suc csuc 4051
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-13 1385  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  ax-pr 3918  ax-un 4120  ax-setind 4204
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  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  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-tr 3829  df-eprel 4000  df-iso 4008  df-iord 4052  df-on 4054  df-suc 4057
This theorem is referenced by: (None)
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