Theorem ordpwsucexmid 4246

Description: The subset in ordpwsucss 4243 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 3908	. . . . 5 ⊢ ∅ ∈ 𝒫 {z ∈ {∅} ∣ φ}
2		0elon 4095	. . . . 5 ⊢ ∅ ∈ On
3		elin 3120	. . . . 5 ⊢ (∅ ∈ (𝒫 {z ∈ {∅} ∣ φ} ∩ On) ↔ (∅ ∈ 𝒫 {z ∈ {∅} ∣ φ} ∧ ∅ ∈ On))
4	1, 2, 3	mpbir2an 848	. . . 4 ⊢ ∅ ∈ (𝒫 {z ∈ {∅} ∣ φ} ∩ On)
5		ordtriexmidlem 4208	. . . . 5 ⊢ {z ∈ {∅} ∣ φ} ∈ On
6		suceq 4105	. . . . . . 7 ⊢ (x = {z ∈ {∅} ∣ φ} → suc x = suc {z ∈ {∅} ∣ φ})
7		pweq 3354	. . . . . . . 8 ⊢ (x = {z ∈ {∅} ∣ φ} → 𝒫 x = 𝒫 {z ∈ {∅} ∣ φ})
8	7	ineq1d 3131	. . . . . . 7 ⊢ (x = {z ∈ {∅} ∣ φ} → (𝒫 x ∩ On) = (𝒫 {z ∈ {∅} ∣ φ} ∩ On))
9	6, 8	eqeq12d 2051	. . . . . 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 2622	. . . . 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 2110	. . 3 ⊢ ∅ ∈ suc {z ∈ {∅} ∣ φ}
14		elsuci 4106	. . 3 ⊢ (∅ ∈ suc {z ∈ {∅} ∣ φ} → (∅ ∈ {z ∈ {∅} ∣ φ} ∨ ∅ = {z ∈ {∅} ∣ φ}))
15	13, 14	ax-mp 7	. 2 ⊢ (∅ ∈ {z ∈ {∅} ∣ φ} ∨ ∅ = {z ∈ {∅} ∣ φ})
16		0ex 3875	. . . . . 6 ⊢ ∅ ∈ V
17	16	snid 3394	. . . . 5 ⊢ ∅ ∈ {∅}
18		biidd 161	. . . . . 6 ⊢ (z = ∅ → (φ ↔ φ))
19	18	elrab3 2693	. . . . 5 ⊢ (∅ ∈ {∅} → (∅ ∈ {z ∈ {∅} ∣ φ} ↔ φ))
20	17, 19	ax-mp 7	. . . 4 ⊢ (∅ ∈ {z ∈ {∅} ∣ φ} ↔ φ)
21	20	biimpi 113	. . 3 ⊢ (∅ ∈ {z ∈ {∅} ∣ φ} → φ)
22		ordtriexmidlem2 4209	. . . 4 ⊢ ({z ∈ {∅} ∣ φ} = ∅ → ¬ φ)
23	22	eqcoms 2040	. . 3 ⊢ (∅ = {z ∈ {∅} ∣ φ} → ¬ φ)
24	21, 23	orim12i 675	. 2 ⊢ ((∅ ∈ {z ∈ {∅} ∣ φ} ∨ ∅ = {z ∈ {∅} ∣ φ}) → (φ ∨ ¬ φ))
25	15, 24	ax-mp 7	1 ⊢ (φ ∨ ¬ φ)

Syntax hints:  ¬ wn 3   ↔ wb 98   ∨ wo 628   = wceq 1242   ∈ wcel 1390  ∀wral 2300  {crab 2304   ∩ cin 2910  ∅c0 3218  𝒫 cpw 3351  {csn 3367  Oncon0 4066  suc csuc 4068

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071  df-suc 4074

This theorem is referenced by: (None)