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Theorem ordpwsucexmid 4218
Description: The subset in ordpwsucss 4215 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 3880 . . . . 5 𝒫 {z {∅} ∣ φ}
2 0elon 4067 . . . . 5 On
3 elin 3094 . . . . 5 (∅ (𝒫 {z {∅} ∣ φ} ∩ On) ↔ (∅ 𝒫 {z {∅} ∣ φ} On))
41, 2, 3mpbir2an 831 . . . 4 (𝒫 {z {∅} ∣ φ} ∩ On)
5 ordtriexmidlem 4180 . . . . 5 {z {∅} ∣ φ} On
6 suceq 4077 . . . . . . 7 (x = {z {∅} ∣ φ} → suc x = suc {z {∅} ∣ φ})
7 pweq 3326 . . . . . . . 8 (x = {z {∅} ∣ φ} → 𝒫 x = 𝒫 {z {∅} ∣ φ})
87ineq1d 3105 . . . . . . 7 (x = {z {∅} ∣ φ} → (𝒫 x ∩ On) = (𝒫 {z {∅} ∣ φ} ∩ On))
96, 8eqeq12d 2027 . . . . . 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 2596 . . . . 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 2086 . . 3 suc {z {∅} ∣ φ}
14 elsuci 4078 . . 3 (∅ suc {z {∅} ∣ φ} → (∅ {z {∅} ∣ φ} ∅ = {z {∅} ∣ φ}))
1513, 14ax-mp 7 . 2 (∅ {z {∅} ∣ φ} ∅ = {z {∅} ∣ φ})
16 0ex 3847 . . . . . 6 V
1716snid 3366 . . . . 5 {∅}
18 biidd 161 . . . . . 6 (z = ∅ → (φφ))
1918elrab3 2667 . . . . 5 (∅ {∅} → (∅ {z {∅} ∣ φ} ↔ φ))
2017, 19ax-mp 7 . . . 4 (∅ {z {∅} ∣ φ} ↔ φ)
2120biimpi 113 . . 3 (∅ {z {∅} ∣ φ} → φ)
22 ordtriexmidlem2 4181 . . . 4 ({z {∅} ∣ φ} = ∅ → ¬ φ)
2322eqcoms 2016 . . 3 (∅ = {z {∅} ∣ φ} → ¬ φ)
2421, 23orim12i 660 . 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 613   = wceq 1223   wcel 1366  wral 2275  {crab 2279  cin 2884  c0 3192  𝒫 cpw 3323  {csn 3339  Oncon0 4038  suc csuc 4040
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 529  ax-in2 530  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838  ax-nul 3846  ax-pow 3890
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-rab 2284  df-v 2528  df-dif 2888  df-un 2890  df-in 2892  df-ss 2899  df-nul 3193  df-pw 3325  df-sn 3345  df-uni 3544  df-tr 3818  df-iord 4041  df-on 4043  df-suc 4046
This theorem is referenced by: (None)
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