Theorem ordpwsucexmid 4414

Description: The subset in ordpwsucss 4411 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 4020	. . . . 5 ⊢ ∅ ∈ 𝒫 {𝑧 ∈ {∅} ∣ 𝜑}
2		0elon 4243	. . . . 5 ⊢ ∅ ∈ On
3		elin 3198	. . . . 5 ⊢ (∅ ∈ (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On) ↔ (∅ ∈ 𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∧ ∅ ∈ On))
4	1, 2, 3	mpbir2an 891	. . . 4 ⊢ ∅ ∈ (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On)
5		ordtriexmidlem 4364	. . . . 5 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
6		suceq 4253	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑧 ∈ {∅} ∣ 𝜑})
7		pweq 3452	. . . . . . . 8 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → 𝒫 𝑥 = 𝒫 {𝑧 ∈ {∅} ∣ 𝜑})
8	7	ineq1d 3215	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝒫 𝑥 ∩ On) = (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On))
9	6, 8	eqeq12d 2109	. . . . . 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 2708	. . . . 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 2170	. . 3 ⊢ ∅ ∈ suc {𝑧 ∈ {∅} ∣ 𝜑}
14		elsuci 4254	. . 3 ⊢ (∅ ∈ suc {𝑧 ∈ {∅} ∣ 𝜑} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑧 ∈ {∅} ∣ 𝜑}))
15	13, 14	ax-mp 7	. 2 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑧 ∈ {∅} ∣ 𝜑})
16		0ex 3987	. . . . . 6 ⊢ ∅ ∈ V
17	16	snid 3495	. . . . 5 ⊢ ∅ ∈ {∅}
18		biidd 171	. . . . . 6 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
19	18	elrab3 2786	. . . . 5 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
20	17, 19	ax-mp 7	. . . 4 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
21	20	biimpi 119	. . 3 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
22		ordtriexmidlem2 4365	. . . 4 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
23	22	eqcoms 2098	. . 3 ⊢ (∅ = {𝑧 ∈ {∅} ∣ 𝜑} → ¬ 𝜑)
24	21, 23	orim12i 714	. 2 ⊢ ((∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑧 ∈ {∅} ∣ 𝜑}) → (𝜑 ∨ ¬ 𝜑))
25	15, 24	ax-mp 7	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 104   ∨ wo 667   = wceq 1296   ∈ wcel 1445  ∀wral 2370  {crab 2374   ∩ cin 3012  ∅c0 3302  𝒫 cpw 3449  {csn 3466  Oncon0 4214  suc csuc 4216

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-uni 3676  df-tr 3959  df-iord 4217  df-on 4219  df-suc 4222

This theorem is referenced by: (None)