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Theorem ordpwsucexmid 4246
 Description: The subset in ordpwsucss 4243 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 3908 . . . . 5 𝒫 {z {∅} ∣ φ}
2 0elon 4095 . . . . 5 On
3 elin 3120 . . . . 5 (∅ (𝒫 {z {∅} ∣ φ} ∩ On) ↔ (∅ 𝒫 {z {∅} ∣ φ} On))
41, 2, 3mpbir2an 848 . . . 4 (𝒫 {z {∅} ∣ φ} ∩ On)
5 ordtriexmidlem 4208 . . . . 5 {z {∅} ∣ φ} On
6 suceq 4105 . . . . . . 7 (x = {z {∅} ∣ φ} → suc x = suc {z {∅} ∣ φ})
7 pweq 3354 . . . . . . . 8 (x = {z {∅} ∣ φ} → 𝒫 x = 𝒫 {z {∅} ∣ φ})
87ineq1d 3131 . . . . . . 7 (x = {z {∅} ∣ φ} → (𝒫 x ∩ On) = (𝒫 {z {∅} ∣ φ} ∩ On))
96, 8eqeq12d 2051 . . . . . 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 2622 . . . . 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 2110 . . 3 suc {z {∅} ∣ φ}
14 elsuci 4106 . . 3 (∅ suc {z {∅} ∣ φ} → (∅ {z {∅} ∣ φ} ∅ = {z {∅} ∣ φ}))
1513, 14ax-mp 7 . 2 (∅ {z {∅} ∣ φ} ∅ = {z {∅} ∣ φ})
16 0ex 3875 . . . . . 6 V
1716snid 3394 . . . . 5 {∅}
18 biidd 161 . . . . . 6 (z = ∅ → (φφ))
1918elrab3 2693 . . . . 5 (∅ {∅} → (∅ {z {∅} ∣ φ} ↔ φ))
2017, 19ax-mp 7 . . . 4 (∅ {z {∅} ∣ φ} ↔ φ)
2120biimpi 113 . . 3 (∅ {z {∅} ∣ φ} → φ)
22 ordtriexmidlem2 4209 . . . 4 ({z {∅} ∣ φ} = ∅ → ¬ φ)
2322eqcoms 2040 . . 3 (∅ = {z {∅} ∣ φ} → ¬ φ)
2421, 23orim12i 675 . 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 628   = wceq 1242   ∈ wcel 1390  ∀wral 2300  {crab 2304   ∩ cin 2910  ∅c0 3218  𝒫 cpw 3351  {csn 3367  Oncon0 4066  suc csuc 4068 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-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071  df-suc 4074 This theorem is referenced by: (None)
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