Theorem ordpwsucexmid 4218

Description: The subset in ordpwsucss 4215 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 3880	. . . . 5 ⊢ ∅ ∈ 𝒫 {z ∈ {∅} ∣ φ}
2		0elon 4067	. . . . 5 ⊢ ∅ ∈ On
3		elin 3094	. . . . 5 ⊢ (∅ ∈ (𝒫 {z ∈ {∅} ∣ φ} ∩ On) ↔ (∅ ∈ 𝒫 {z ∈ {∅} ∣ φ} ∧ ∅ ∈ On))
4	1, 2, 3	mpbir2an 831	. . . 4 ⊢ ∅ ∈ (𝒫 {z ∈ {∅} ∣ φ} ∩ On)
5		ordtriexmidlem 4180	. . . . 5 ⊢ {z ∈ {∅} ∣ φ} ∈ On
6		suceq 4077	. . . . . . 7 ⊢ (x = {z ∈ {∅} ∣ φ} → suc x = suc {z ∈ {∅} ∣ φ})
7		pweq 3326	. . . . . . . 8 ⊢ (x = {z ∈ {∅} ∣ φ} → 𝒫 x = 𝒫 {z ∈ {∅} ∣ φ})
8	7	ineq1d 3105	. . . . . . 7 ⊢ (x = {z ∈ {∅} ∣ φ} → (𝒫 x ∩ On) = (𝒫 {z ∈ {∅} ∣ φ} ∩ On))
9	6, 8	eqeq12d 2027	. . . . . 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 2596	. . . . 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 2086	. . 3 ⊢ ∅ ∈ suc {z ∈ {∅} ∣ φ}
14		elsuci 4078	. . 3 ⊢ (∅ ∈ suc {z ∈ {∅} ∣ φ} → (∅ ∈ {z ∈ {∅} ∣ φ} ∨ ∅ = {z ∈ {∅} ∣ φ}))
15	13, 14	ax-mp 7	. 2 ⊢ (∅ ∈ {z ∈ {∅} ∣ φ} ∨ ∅ = {z ∈ {∅} ∣ φ})
16		0ex 3847	. . . . . 6 ⊢ ∅ ∈ V
17	16	snid 3366	. . . . 5 ⊢ ∅ ∈ {∅}
18		biidd 161	. . . . . 6 ⊢ (z = ∅ → (φ ↔ φ))
19	18	elrab3 2667	. . . . 5 ⊢ (∅ ∈ {∅} → (∅ ∈ {z ∈ {∅} ∣ φ} ↔ φ))
20	17, 19	ax-mp 7	. . . 4 ⊢ (∅ ∈ {z ∈ {∅} ∣ φ} ↔ φ)
21	20	biimpi 113	. . 3 ⊢ (∅ ∈ {z ∈ {∅} ∣ φ} → φ)
22		ordtriexmidlem2 4181	. . . 4 ⊢ ({z ∈ {∅} ∣ φ} = ∅ → ¬ φ)
23	22	eqcoms 2016	. . 3 ⊢ (∅ = {z ∈ {∅} ∣ φ} → ¬ φ)
24	21, 23	orim12i 660	. 2 ⊢ ((∅ ∈ {z ∈ {∅} ∣ φ} ∨ ∅ = {z ∈ {∅} ∣ φ}) → (φ ∨ ¬ φ))
25	15, 24	ax-mp 7	1 ⊢ (φ ∨ ¬ φ)

Syntax hints:  ¬ wn 3   ↔ wb 98   ∨ wo 613   = wceq 1223   ∈ wcel 1366  ∀wral 2275  {crab 2279   ∩ cin 2884  ∅c0 3192  𝒫 cpw 3323  {csn 3339  Oncon0 4038  suc csuc 4040

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 529  ax-in2 530  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838  ax-nul 3846  ax-pow 3890

This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-rab 2284  df-v 2528  df-dif 2888  df-un 2890  df-in 2892  df-ss 2899  df-nul 3193  df-pw 3325  df-sn 3345  df-uni 3544  df-tr 3818  df-iord 4041  df-on 4043  df-suc 4046

This theorem is referenced by: (None)