Theorem exmidpweq 6965

Description: Excluded middle is equivalent to the power set of 1_o being 2_o. (Contributed by Jim Kingdon, 28-Jul-2024.)

Assertion

Ref	Expression
exmidpweq	⊢ (EXMID ↔ 𝒫 1o = 2o)

Proof of Theorem exmidpweq

Step	Hyp	Ref	Expression
1		exmid01 4227	. . . . . . . 8 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
2	1	biimpi 120	. . . . . . 7 ⊢ (EXMID → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
3	2	19.21bi 1569	. . . . . 6 ⊢ (EXMID → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
4		df1o2 6482	. . . . . . . . 9 ⊢ 1o = {∅}
5	4	pweqi 3605	. . . . . . . 8 ⊢ 𝒫 1o = 𝒫 {∅}
6	5	eleq2i 2260	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 1o ↔ 𝑥 ∈ 𝒫 {∅})
7		velpw 3608	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 {∅} ↔ 𝑥 ⊆ {∅})
8	6, 7	bitri 184	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 1o ↔ 𝑥 ⊆ {∅})
9		vex 2763	. . . . . . 7 ⊢ 𝑥 ∈ V
10	9	elpr 3639	. . . . . 6 ⊢ (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
11	3, 8, 10	3imtr4g 205	. . . . 5 ⊢ (EXMID → (𝑥 ∈ 𝒫 1o → 𝑥 ∈ {∅, {∅}}))
12	11	ssrdv 3185	. . . 4 ⊢ (EXMID → 𝒫 1o ⊆ {∅, {∅}})
13		pwpw0ss 3830	. . . . . 6 ⊢ {∅, {∅}} ⊆ 𝒫 {∅}
14	13, 5	sseqtrri 3214	. . . . 5 ⊢ {∅, {∅}} ⊆ 𝒫 1o
15	14	a1i 9	. . . 4 ⊢ (EXMID → {∅, {∅}} ⊆ 𝒫 1o)
16	12, 15	eqssd 3196	. . 3 ⊢ (EXMID → 𝒫 1o = {∅, {∅}})
17		df2o2 6484	. . 3 ⊢ 2o = {∅, {∅}}
18	16, 17	eqtr4di 2244	. 2 ⊢ (EXMID → 𝒫 1o = 2o)
19		simpr 110	. . . . . . . . 9 ⊢ ((𝒫 1o = 2o ∧ 𝑥 ⊆ {∅}) → 𝑥 ⊆ {∅})
20	19, 7	sylibr 134	. . . . . . . 8 ⊢ ((𝒫 1o = 2o ∧ 𝑥 ⊆ {∅}) → 𝑥 ∈ 𝒫 {∅})
21	20, 5	eleqtrrdi 2287	. . . . . . 7 ⊢ ((𝒫 1o = 2o ∧ 𝑥 ⊆ {∅}) → 𝑥 ∈ 𝒫 1o)
22		simpl 109	. . . . . . . 8 ⊢ ((𝒫 1o = 2o ∧ 𝑥 ⊆ {∅}) → 𝒫 1o = 2o)
23	22, 17	eqtrdi 2242	. . . . . . 7 ⊢ ((𝒫 1o = 2o ∧ 𝑥 ⊆ {∅}) → 𝒫 1o = {∅, {∅}})
24	21, 23	eleqtrd 2272	. . . . . 6 ⊢ ((𝒫 1o = 2o ∧ 𝑥 ⊆ {∅}) → 𝑥 ∈ {∅, {∅}})
25	24, 10	sylib 122	. . . . 5 ⊢ ((𝒫 1o = 2o ∧ 𝑥 ⊆ {∅}) → (𝑥 = ∅ ∨ 𝑥 = {∅}))
26	25	ex 115	. . . 4 ⊢ (𝒫 1o = 2o → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
27	26	alrimiv 1885	. . 3 ⊢ (𝒫 1o = 2o → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
28	27, 1	sylibr 134	. 2 ⊢ (𝒫 1o = 2o → EXMID)
29	18, 28	impbii 126	1 ⊢ (EXMID ↔ 𝒫 1o = 2o)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  ∀wal 1362   = wceq 1364   ∈ wcel 2164   ⊆ wss 3153  ∅c0 3446  𝒫 cpw 3601  {csn 3618  {cpr 3619  EXMIDwem 4223  1_oc1o 6462  2_oc2o 6463

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-nul 4155

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-exmid 4224  df-suc 4402  df-1o 6469  df-2o 6470

This theorem is referenced by:  pw1fin  6966  pw1nel3  7291  3nsssucpw1  7296  onntri35  7297