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Theorem ordpwsucexmid 4199
Description: The subset in ordpwsucss 4196 cannot be equality. That is, strengthening it to equality implies excluded middle. (Contributed by Jim Kingdon, 30-Jul-2019.)
Hypothesis
Ref Expression
ordpwsucexmid.1 x On suc x = (𝒫 x ∩ On)
Assertion
Ref Expression
ordpwsucexmid (φ ¬ φ)
Distinct variable group:   φ,x

Proof of Theorem ordpwsucexmid
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 0elpw 3869 . . . . 5 𝒫 {z {∅} ∣ φ}
2 0elon 4052 . . . . 5 On
3 elin 3104 . . . . 5 (∅ (𝒫 {z {∅} ∣ φ} ∩ On) ↔ (∅ 𝒫 {z {∅} ∣ φ} On))
41, 2, 3mpbir2an 837 . . . 4 (𝒫 {z {∅} ∣ φ} ∩ On)
5 ordtriexmidlem 4165 . . . . 5 {z {∅} ∣ φ} On
6 suceq 4062 . . . . . . 7 (x = {z {∅} ∣ φ} → suc x = suc {z {∅} ∣ φ})
7 pweq 3314 . . . . . . . 8 (x = {z {∅} ∣ φ} → 𝒫 x = 𝒫 {z {∅} ∣ φ})
87ineq1d 3115 . . . . . . 7 (x = {z {∅} ∣ φ} → (𝒫 x ∩ On) = (𝒫 {z {∅} ∣ φ} ∩ On))
96, 8eqeq12d 2036 . . . . . 6 (x = {z {∅} ∣ φ} → (suc x = (𝒫 x ∩ On) ↔ suc {z {∅} ∣ φ} = (𝒫 {z {∅} ∣ φ} ∩ On)))
10 ordpwsucexmid.1 . . . . . 6 x On suc x = (𝒫 x ∩ On)
119, 10vtoclri 2603 . . . . 5 ({z {∅} ∣ φ} On → suc {z {∅} ∣ φ} = (𝒫 {z {∅} ∣ φ} ∩ On))
125, 11ax-mp 7 . . . 4 suc {z {∅} ∣ φ} = (𝒫 {z {∅} ∣ φ} ∩ On)
134, 12eleqtrri 2095 . . 3 suc {z {∅} ∣ φ}
14 elsuci 4063 . . 3 (∅ suc {z {∅} ∣ φ} → (∅ {z {∅} ∣ φ} ∅ = {z {∅} ∣ φ}))
1513, 14ax-mp 7 . 2 (∅ {z {∅} ∣ φ} ∅ = {z {∅} ∣ φ})
16 0ex 3836 . . . . . 6 V
1716snid 3354 . . . . 5 {∅}
18 biidd 161 . . . . . 6 (z = ∅ → (φφ))
1918elrab3 2674 . . . . 5 (∅ {∅} → (∅ {z {∅} ∣ φ} ↔ φ))
2017, 19ax-mp 7 . . . 4 (∅ {z {∅} ∣ φ} ↔ φ)
2120biimpi 113 . . 3 (∅ {z {∅} ∣ φ} → φ)
22 ordtriexmidlem2 4166 . . . 4 ({z {∅} ∣ φ} = ∅ → ¬ φ)
2322eqcoms 2025 . . 3 (∅ = {z {∅} ∣ φ} → ¬ φ)
2421, 23orim12i 664 . 2 ((∅ {z {∅} ∣ φ} ∅ = {z {∅} ∣ φ}) → (φ ¬ φ))
2515, 24ax-mp 7 1 (φ ¬ φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 98   wo 616   = wceq 1373   wcel 1375  wral 2282  {crab 2286  cin 2894  c0 3202  𝒫 cpw 3311  {csn 3327  Oncon0 4024  suc csuc 4026
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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-sep 3827  ax-nul 3835  ax-pow 3879
This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-rab 2291  df-v 2535  df-dif 2898  df-un 2900  df-in 2902  df-ss 2909  df-nul 3203  df-pw 3313  df-sn 3333  df-uni 3533  df-tr 3807  df-iord 4027  df-on 4028  df-suc 4031
This theorem is referenced by: (None)
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