Theorem ordpwsucexmid 4570

Description: The subset in ordpwsucss 4567 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 4165	. . . . 5 ⊢ ∅ ∈ 𝒫 {𝑧 ∈ {∅} ∣ 𝜑}
2		0elon 4393	. . . . 5 ⊢ ∅ ∈ On
3		elin 3319	. . . . 5 ⊢ (∅ ∈ (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On) ↔ (∅ ∈ 𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∧ ∅ ∈ On))
4	1, 2, 3	mpbir2an 942	. . . 4 ⊢ ∅ ∈ (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On)
5		ordtriexmidlem 4519	. . . . 5 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
6		suceq 4403	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → suc 𝑥 = suc {𝑧 ∈ {∅} ∣ 𝜑})
7		pweq 3579	. . . . . . . 8 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → 𝒫 𝑥 = 𝒫 {𝑧 ∈ {∅} ∣ 𝜑})
8	7	ineq1d 3336	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝒫 𝑥 ∩ On) = (𝒫 {𝑧 ∈ {∅} ∣ 𝜑} ∩ On))
9	6, 8	eqeq12d 2192	. . . . . 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 2813	. . . . 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 2253	. . 3 ⊢ ∅ ∈ suc {𝑧 ∈ {∅} ∣ 𝜑}
14		elsuci 4404	. . 3 ⊢ (∅ ∈ suc {𝑧 ∈ {∅} ∣ 𝜑} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑧 ∈ {∅} ∣ 𝜑}))
15	13, 14	ax-mp 5	. 2 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑧 ∈ {∅} ∣ 𝜑})
16		0ex 4131	. . . . . 6 ⊢ ∅ ∈ V
17	16	snid 3624	. . . . 5 ⊢ ∅ ∈ {∅}
18		biidd 172	. . . . . 6 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
19	18	elrab3 2895	. . . . 5 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
20	17, 19	ax-mp 5	. . . 4 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
21	20	biimpi 120	. . 3 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
22		ordtriexmidlem2 4520	. . . 4 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
23	22	eqcoms 2180	. . 3 ⊢ (∅ = {𝑧 ∈ {∅} ∣ 𝜑} → ¬ 𝜑)
24	21, 23	orim12i 759	. 2 ⊢ ((∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ∨ ∅ = {𝑧 ∈ {∅} ∣ 𝜑}) → (𝜑 ∨ ¬ 𝜑))
25	15, 24	ax-mp 5	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 105   ∨ wo 708   = wceq 1353   ∈ wcel 2148  ∀wral 2455  {crab 2459   ∩ cin 3129  ∅c0 3423  𝒫 cpw 3576  {csn 3593  Oncon0 4364  suc csuc 4366

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-uni 3811  df-tr 4103  df-iord 4367  df-on 4369  df-suc 4372

This theorem is referenced by: (None)