Theorem ordpwsucexmid 4279

Description: The subset in ordpwsucss 4276 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 3914	. . . . 5 ⊢ ∅ ∈ 𝒫 {𝑧 ∈ {∅} ∣ 𝜑}
2		0elon 4116	. . . . 5 ⊢ ∅ ∈ On
3		elin 3123	. . . . 5 ⊢ (∅ ∈ (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On) ↔ (∅ ∈ 𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∧ ∅ ∈ On))
4	1, 2, 3	mpbir2an 849	. . . 4 ⊢ ∅ ∈ (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On)
5		ordtriexmidlem 4232	. . . . 5 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
6		suceq 4126	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑧 ∈ {∅} ∣ 𝜑})
7		pweq 3359	. . . . . . . 8 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → 𝒫 𝑥 = 𝒫 {𝑧 ∈ {∅} ∣ 𝜑})
8	7	ineq1d 3134	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝒫 𝑥 ∩ On) = (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On))
9	6, 8	eqeq12d 2054	. . . . . 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 2625	. . . . 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 2113	. . 3 ⊢ ∅ ∈ suc {𝑧 ∈ {∅} ∣ 𝜑}
14		elsuci 4127	. . 3 ⊢ (∅ ∈ suc {𝑧 ∈ {∅} ∣ 𝜑} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑧 ∈ {∅} ∣ 𝜑}))
15	13, 14	ax-mp 7	. 2 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑧 ∈ {∅} ∣ 𝜑})
16		0ex 3881	. . . . . 6 ⊢ ∅ ∈ V
17	16	snid 3399	. . . . 5 ⊢ ∅ ∈ {∅}
18		biidd 161	. . . . . 6 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
19	18	elrab3 2696	. . . . 5 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
20	17, 19	ax-mp 7	. . . 4 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
21	20	biimpi 113	. . 3 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
22		ordtriexmidlem2 4233	. . . 4 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
23	22	eqcoms 2043	. . 3 ⊢ (∅ = {𝑧 ∈ {∅} ∣ 𝜑} → ¬ 𝜑)
24	21, 23	orim12i 676	. 2 ⊢ ((∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑧 ∈ {∅} ∣ 𝜑}) → (𝜑 ∨ ¬ 𝜑))
25	15, 24	ax-mp 7	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 98   ∨ wo 629   = wceq 1243   ∈ wcel 1393  ∀wral 2303  {crab 2307   ∩ cin 2913  ∅c0 3221  𝒫 cpw 3356  {csn 3372  Oncon0 4087  suc csuc 4089

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-nul 3880  ax-pow 3924

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-rab 2312  df-v 2556  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-uni 3578  df-tr 3852  df-iord 4090  df-on 4092  df-suc 4095

This theorem is referenced by: (None)