Theorem ordpwsucexmid 4670

Description: The subset in ordpwsucss 4667 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 4256	. . . . 5 ⊢ ∅ ∈ 𝒫 {𝑧 ∈ {∅} ∣ 𝜑}
2		0elon 4491	. . . . 5 ⊢ ∅ ∈ On
3		elin 3389	. . . . 5 ⊢ (∅ ∈ (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On) ↔ (∅ ∈ 𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∧ ∅ ∈ On))
4	1, 2, 3	mpbir2an 950	. . . 4 ⊢ ∅ ∈ (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On)
5		ordtriexmidlem 4619	. . . . 5 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
6		suceq 4501	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑧 ∈ {∅} ∣ 𝜑})
7		pweq 3656	. . . . . . . 8 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → 𝒫 𝑥 = 𝒫 {𝑧 ∈ {∅} ∣ 𝜑})
8	7	ineq1d 3406	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝒫 𝑥 ∩ On) = (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On))
9	6, 8	eqeq12d 2245	. . . . . 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 2880	. . . . 5 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ On → suc {𝑧 ∈ {∅} ∣ 𝜑} = (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On))
12	5, 11	ax-mp 5	. . . 4 ⊢ suc {𝑧 ∈ {∅} ∣ 𝜑} = (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On)
13	4, 12	eleqtrri 2306	. . 3 ⊢ ∅ ∈ suc {𝑧 ∈ {∅} ∣ 𝜑}
14		elsuci 4502	. . 3 ⊢ (∅ ∈ suc {𝑧 ∈ {∅} ∣ 𝜑} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑧 ∈ {∅} ∣ 𝜑}))
15	13, 14	ax-mp 5	. 2 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑧 ∈ {∅} ∣ 𝜑})
16		0ex 4217	. . . . . 6 ⊢ ∅ ∈ V
17	16	snid 3701	. . . . 5 ⊢ ∅ ∈ {∅}
18		biidd 172	. . . . . 6 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
19	18	elrab3 2962	. . . . 5 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
20	17, 19	ax-mp 5	. . . 4 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
21	20	biimpi 120	. . 3 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
22		ordtriexmidlem2 4620	. . . 4 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
23	22	eqcoms 2233	. . 3 ⊢ (∅ = {𝑧 ∈ {∅} ∣ 𝜑} → ¬ 𝜑)
24	21, 23	orim12i 766	. 2 ⊢ ((∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑧 ∈ {∅} ∣ 𝜑}) → (𝜑 ∨ ¬ 𝜑))
25	15, 24	ax-mp 5	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 105   ∨ wo 715   = wceq 1397   ∈ wcel 2201  ∀wral 2509  {crab 2513   ∩ cin 3198  ∅c0 3493  𝒫 cpw 3653  {csn 3670  Oncon0 4462  suc csuc 4464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-nul 4216  ax-pow 4266

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-rab 2518  df-v 2803  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-uni 3895  df-tr 4189  df-iord 4465  df-on 4467  df-suc 4470

This theorem is referenced by: (None)