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Theorem ordpwsucexmid 4279
Description: The subset in ordpwsucss 4276 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 3914 . . . . 5 ∅ ∈ 𝒫 {𝑧 ∈ {∅} ∣ 𝜑}
2 0elon 4116 . . . . 5 ∅ ∈ On
3 elin 3123 . . . . 5 (∅ ∈ (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On) ↔ (∅ ∈ 𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∧ ∅ ∈ On))
41, 2, 3mpbir2an 849 . . . 4 ∅ ∈ (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On)
5 ordtriexmidlem 4232 . . . . 5 {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
6 suceq 4126 . . . . . . 7 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑧 ∈ {∅} ∣ 𝜑})
7 pweq 3359 . . . . . . . 8 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → 𝒫 𝑥 = 𝒫 {𝑧 ∈ {∅} ∣ 𝜑})
87ineq1d 3134 . . . . . . 7 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝒫 𝑥 ∩ On) = (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On))
96, 8eqeq12d 2054 . . . . . 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 2625 . . . . 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 2113 . . 3 ∅ ∈ suc {𝑧 ∈ {∅} ∣ 𝜑}
14 elsuci 4127 . . 3 (∅ ∈ suc {𝑧 ∈ {∅} ∣ 𝜑} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑧 ∈ {∅} ∣ 𝜑}))
1513, 14ax-mp 7 . 2 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑧 ∈ {∅} ∣ 𝜑})
16 0ex 3881 . . . . . 6 ∅ ∈ V
1716snid 3399 . . . . 5 ∅ ∈ {∅}
18 biidd 161 . . . . . 6 (𝑧 = ∅ → (𝜑𝜑))
1918elrab3 2696 . . . . 5 (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
2017, 19ax-mp 7 . . . 4 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
2120biimpi 113 . . 3 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
22 ordtriexmidlem2 4233 . . . 4 ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
2322eqcoms 2043 . . 3 (∅ = {𝑧 ∈ {∅} ∣ 𝜑} → ¬ 𝜑)
2421, 23orim12i 676 . 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 629   = wceq 1243  wcel 1393  wral 2303  {crab 2307  cin 2913  c0 3221  𝒫 cpw 3356  {csn 3372  Oncon0 4087  suc csuc 4089
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-nul 3880  ax-pow 3924
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-rab 2312  df-v 2556  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-uni 3578  df-tr 3852  df-iord 4090  df-on 4092  df-suc 4095
This theorem is referenced by: (None)
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