Theorem ordpwsucexmid 4352

Description: The subset in ordpwsucss 4349 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.

Step	Hyp	Ref	Expression
1		0elpw 3967	. . . . 5 ⊢ ∅ ∈ 𝒫 {𝑧 ∈ {∅} ∣ 𝜑}
2		0elon 4186	. . . . 5 ⊢ ∅ ∈ On
3		elin 3169	. . . . 5 ⊢ (∅ ∈ (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On) ↔ (∅ ∈ 𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∧ ∅ ∈ On))
4	1, 2, 3	mpbir2an 886	. . . 4 ⊢ ∅ ∈ (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On)
5		ordtriexmidlem 4302	. . . . 5 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
6		suceq 4196	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑧 ∈ {∅} ∣ 𝜑})
7		pweq 3412	. . . . . . . 8 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → 𝒫 𝑥 = 𝒫 {𝑧 ∈ {∅} ∣ 𝜑})
8	7	ineq1d 3186	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝒫 𝑥 ∩ On) = (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On))
9	6, 8	eqeq12d 2099	. . . . . 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)
11	9, 10	vtoclri 2686	. . . . 5 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ On → suc {𝑧 ∈ {∅} ∣ 𝜑} = (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On))
12	5, 11	ax-mp 7	. . . 4 ⊢ suc {𝑧 ∈ {∅} ∣ 𝜑} = (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On)
13	4, 12	eleqtrri 2160	. . 3 ⊢ ∅ ∈ suc {𝑧 ∈ {∅} ∣ 𝜑}
14		elsuci 4197	. . 3 ⊢ (∅ ∈ suc {𝑧 ∈ {∅} ∣ 𝜑} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑧 ∈ {∅} ∣ 𝜑}))
15	13, 14	ax-mp 7	. 2 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑧 ∈ {∅} ∣ 𝜑})
16		0ex 3934	. . . . . 6 ⊢ ∅ ∈ V
17	16	snid 3452	. . . . 5 ⊢ ∅ ∈ {∅}
18		biidd 170	. . . . . 6 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
19	18	elrab3 2762	. . . . 5 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
20	17, 19	ax-mp 7	. . . 4 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
21	20	biimpi 118	. . 3 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
22		ordtriexmidlem2 4303	. . . 4 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
23	22	eqcoms 2088	. . 3 ⊢ (∅ = {𝑧 ∈ {∅} ∣ 𝜑} → ¬ 𝜑)
24	21, 23	orim12i 709	. 2 ⊢ ((∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑧 ∈ {∅} ∣ 𝜑}) → (𝜑 ∨ ¬ 𝜑))
25	15, 24	ax-mp 7	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 103   ∨ wo 662   = wceq 1287   ∈ wcel 1436  ∀wral 2355  {crab 2359   ∩ cin 2985  ∅c0 3272  𝒫 cpw 3409  {csn 3425  Oncon0 4157  suc csuc 4159

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-nul 3933  ax-pow 3977

This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2616  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-uni 3631  df-tr 3905  df-iord 4160  df-on 4162  df-suc 4165

This theorem is referenced by: (None)