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Theorem ordpwsucexmid 4480
Description: The subset in ordpwsucss 4477 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 4083 . . . . 5 ∅ ∈ 𝒫 {𝑧 ∈ {∅} ∣ 𝜑}
2 0elon 4309 . . . . 5 ∅ ∈ On
3 elin 3254 . . . . 5 (∅ ∈ (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On) ↔ (∅ ∈ 𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∧ ∅ ∈ On))
41, 2, 3mpbir2an 926 . . . 4 ∅ ∈ (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On)
5 ordtriexmidlem 4430 . . . . 5 {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
6 suceq 4319 . . . . . . 7 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑧 ∈ {∅} ∣ 𝜑})
7 pweq 3508 . . . . . . . 8 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → 𝒫 𝑥 = 𝒫 {𝑧 ∈ {∅} ∣ 𝜑})
87ineq1d 3271 . . . . . . 7 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝒫 𝑥 ∩ On) = (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On))
96, 8eqeq12d 2152 . . . . . 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 2756 . . . . 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 2213 . . 3 ∅ ∈ suc {𝑧 ∈ {∅} ∣ 𝜑}
14 elsuci 4320 . . 3 (∅ ∈ suc {𝑧 ∈ {∅} ∣ 𝜑} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑧 ∈ {∅} ∣ 𝜑}))
1513, 14ax-mp 5 . 2 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑧 ∈ {∅} ∣ 𝜑})
16 0ex 4050 . . . . . 6 ∅ ∈ V
1716snid 3551 . . . . 5 ∅ ∈ {∅}
18 biidd 171 . . . . . 6 (𝑧 = ∅ → (𝜑𝜑))
1918elrab3 2836 . . . . 5 (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
2017, 19ax-mp 5 . . . 4 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
2120biimpi 119 . . 3 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
22 ordtriexmidlem2 4431 . . . 4 ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
2322eqcoms 2140 . . 3 (∅ = {𝑧 ∈ {∅} ∣ 𝜑} → ¬ 𝜑)
2421, 23orim12i 748 . 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 104  wo 697   = wceq 1331  wcel 1480  wral 2414  {crab 2418  cin 3065  c0 3358  𝒫 cpw 3505  {csn 3522  Oncon0 4280  suc csuc 4282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-uni 3732  df-tr 4022  df-iord 4283  df-on 4285  df-suc 4288
This theorem is referenced by: (None)
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