Theorem ordpwsucexmid 4228

Description: The subset in ordpwsucss 4225 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 3890	. . . . 5 ⊢ ∅ ∈ 𝒫 {z ∈ {∅} ∣ φ}
2		0elon 4076	. . . . 5 ⊢ ∅ ∈ On
3		elin 3102	. . . . 5 ⊢ (∅ ∈ (𝒫 {z ∈ {∅} ∣ φ} ∩ On) ↔ (∅ ∈ 𝒫 {z ∈ {∅} ∣ φ} ∧ ∅ ∈ On))
4	1, 2, 3	mpbir2an 837	. . . 4 ⊢ ∅ ∈ (𝒫 {z ∈ {∅} ∣ φ} ∩ On)
5		ordtriexmidlem 4190	. . . . 5 ⊢ {z ∈ {∅} ∣ φ} ∈ On
6		suceq 4086	. . . . . . 7 ⊢ (x = {z ∈ {∅} ∣ φ} → suc x = suc {z ∈ {∅} ∣ φ})
7		pweq 3336	. . . . . . . 8 ⊢ (x = {z ∈ {∅} ∣ φ} → 𝒫 x = 𝒫 {z ∈ {∅} ∣ φ})
8	7	ineq1d 3113	. . . . . . 7 ⊢ (x = {z ∈ {∅} ∣ φ} → (𝒫 x ∩ On) = (𝒫 {z ∈ {∅} ∣ φ} ∩ On))
9	6, 8	eqeq12d 2037	. . . . . 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 2604	. . . . 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 2096	. . 3 ⊢ ∅ ∈ suc {z ∈ {∅} ∣ φ}
14		elsuci 4087	. . 3 ⊢ (∅ ∈ suc {z ∈ {∅} ∣ φ} → (∅ ∈ {z ∈ {∅} ∣ φ} ∨ ∅ = {z ∈ {∅} ∣ φ}))
15	13, 14	ax-mp 7	. 2 ⊢ (∅ ∈ {z ∈ {∅} ∣ φ} ∨ ∅ = {z ∈ {∅} ∣ φ})
16		0ex 3857	. . . . . 6 ⊢ ∅ ∈ V
17	16	snid 3376	. . . . 5 ⊢ ∅ ∈ {∅}
18		biidd 161	. . . . . 6 ⊢ (z = ∅ → (φ ↔ φ))
19	18	elrab3 2675	. . . . 5 ⊢ (∅ ∈ {∅} → (∅ ∈ {z ∈ {∅} ∣ φ} ↔ φ))
20	17, 19	ax-mp 7	. . . 4 ⊢ (∅ ∈ {z ∈ {∅} ∣ φ} ↔ φ)
21	20	biimpi 113	. . 3 ⊢ (∅ ∈ {z ∈ {∅} ∣ φ} → φ)
22		ordtriexmidlem2 4191	. . . 4 ⊢ ({z ∈ {∅} ∣ φ} = ∅ → ¬ φ)
23	22	eqcoms 2026	. . 3 ⊢ (∅ = {z ∈ {∅} ∣ φ} → ¬ φ)
24	21, 23	orim12i 663	. 2 ⊢ ((∅ ∈ {z ∈ {∅} ∣ φ} ∨ ∅ = {z ∈ {∅} ∣ φ}) → (φ ∨ ¬ φ))
25	15, 24	ax-mp 7	1 ⊢ (φ ∨ ¬ φ)

Syntax hints:  ¬ wn 3   ↔ wb 98   ∨ wo 616   = wceq 1228   ∈ wcel 1375  ∀wral 2283  {crab 2287   ∩ cin 2892  ∅c0 3200  𝒫 cpw 3333  {csn 3349  Oncon0 4047  suc csuc 4049

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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-sep 3848  ax-nul 3856  ax-pow 3900

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-rab 2292  df-v 2536  df-dif 2896  df-un 2898  df-in 2900  df-ss 2907  df-nul 3201  df-pw 3335  df-sn 3355  df-uni 3554  df-tr 3828  df-iord 4050  df-on 4052  df-suc 4055

This theorem is referenced by: (None)