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Theorem ordpwsucexmid 4587
Description: The subset in ordpwsucss 4584 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 4182 . . . . 5 ∅ ∈ 𝒫 {𝑧 ∈ {∅} ∣ 𝜑}
2 0elon 4410 . . . . 5 ∅ ∈ On
3 elin 3333 . . . . 5 (∅ ∈ (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On) ↔ (∅ ∈ 𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∧ ∅ ∈ On))
41, 2, 3mpbir2an 944 . . . 4 ∅ ∈ (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On)
5 ordtriexmidlem 4536 . . . . 5 {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
6 suceq 4420 . . . . . . 7 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑧 ∈ {∅} ∣ 𝜑})
7 pweq 3593 . . . . . . . 8 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → 𝒫 𝑥 = 𝒫 {𝑧 ∈ {∅} ∣ 𝜑})
87ineq1d 3350 . . . . . . 7 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝒫 𝑥 ∩ On) = (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On))
96, 8eqeq12d 2204 . . . . . 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 2827 . . . . 5 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ On → suc {𝑧 ∈ {∅} ∣ 𝜑} = (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On))
125, 11ax-mp 5 . . . 4 suc {𝑧 ∈ {∅} ∣ 𝜑} = (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On)
134, 12eleqtrri 2265 . . 3 ∅ ∈ suc {𝑧 ∈ {∅} ∣ 𝜑}
14 elsuci 4421 . . 3 (∅ ∈ suc {𝑧 ∈ {∅} ∣ 𝜑} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑧 ∈ {∅} ∣ 𝜑}))
1513, 14ax-mp 5 . 2 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑧 ∈ {∅} ∣ 𝜑})
16 0ex 4145 . . . . . 6 ∅ ∈ V
1716snid 3638 . . . . 5 ∅ ∈ {∅}
18 biidd 172 . . . . . 6 (𝑧 = ∅ → (𝜑𝜑))
1918elrab3 2909 . . . . 5 (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
2017, 19ax-mp 5 . . . 4 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
2120biimpi 120 . . 3 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
22 ordtriexmidlem2 4537 . . . 4 ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
2322eqcoms 2192 . . 3 (∅ = {𝑧 ∈ {∅} ∣ 𝜑} → ¬ 𝜑)
2421, 23orim12i 760 . 2 ((∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑧 ∈ {∅} ∣ 𝜑}) → (𝜑 ∨ ¬ 𝜑))
2515, 24ax-mp 5 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 105  wo 709   = wceq 1364  wcel 2160  wral 2468  {crab 2472  cin 3143  c0 3437  𝒫 cpw 3590  {csn 3607  Oncon0 4381  suc csuc 4383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-uni 3825  df-tr 4117  df-iord 4384  df-on 4386  df-suc 4389
This theorem is referenced by: (None)
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