Theorem exmidpweq 6875

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 4177	. . . . . . . 8 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
2	1	biimpi 119	. . . . . . 7 ⊢ (EXMID → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
3	2	19.21bi 1546	. . . . . 6 ⊢ (EXMID → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
4		df1o2 6397	. . . . . . . . 9 ⊢ 1o = {∅}
5	4	pweqi 3563	. . . . . . . 8 ⊢ 𝒫 1o = 𝒫 {∅}
6	5	eleq2i 2233	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 1o ↔ 𝑥 ∈ 𝒫 {∅})
7		velpw 3566	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 {∅} ↔ 𝑥 ⊆ {∅})
8	6, 7	bitri 183	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 1o ↔ 𝑥 ⊆ {∅})
9		vex 2729	. . . . . . 7 ⊢ 𝑥 ∈ V
10	9	elpr 3597	. . . . . 6 ⊢ (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
11	3, 8, 10	3imtr4g 204	. . . . 5 ⊢ (EXMID → (𝑥 ∈ 𝒫 1o → 𝑥 ∈ {∅, {∅}}))
12	11	ssrdv 3148	. . . 4 ⊢ (EXMID → 𝒫 1o ⊆ {∅, {∅}})
13		pwpw0ss 3784	. . . . . 6 ⊢ {∅, {∅}} ⊆ 𝒫 {∅}
14	13, 5	sseqtrri 3177	. . . . 5 ⊢ {∅, {∅}} ⊆ 𝒫 1o
15	14	a1i 9	. . . 4 ⊢ (EXMID → {∅, {∅}} ⊆ 𝒫 1o)
16	12, 15	eqssd 3159	. . 3 ⊢ (EXMID → 𝒫 1o = {∅, {∅}})
17		df2o2 6399	. . 3 ⊢ 2o = {∅, {∅}}
18	16, 17	eqtr4di 2217	. 2 ⊢ (EXMID → 𝒫 1o = 2o)
19		simpr 109	. . . . . . . . 9 ⊢ ((𝒫 1o = 2o ∧ 𝑥 ⊆ {∅}) → 𝑥 ⊆ {∅})
20	19, 7	sylibr 133	. . . . . . . 8 ⊢ ((𝒫 1o = 2o ∧ 𝑥 ⊆ {∅}) → 𝑥 ∈ 𝒫 {∅})
21	20, 5	eleqtrrdi 2260	. . . . . . 7 ⊢ ((𝒫 1o = 2o ∧ 𝑥 ⊆ {∅}) → 𝑥 ∈ 𝒫 1o)
22		simpl 108	. . . . . . . 8 ⊢ ((𝒫 1o = 2o ∧ 𝑥 ⊆ {∅}) → 𝒫 1o = 2o)
23	22, 17	eqtrdi 2215	. . . . . . 7 ⊢ ((𝒫 1o = 2o ∧ 𝑥 ⊆ {∅}) → 𝒫 1o = {∅, {∅}})
24	21, 23	eleqtrd 2245	. . . . . 6 ⊢ ((𝒫 1o = 2o ∧ 𝑥 ⊆ {∅}) → 𝑥 ∈ {∅, {∅}})
25	24, 10	sylib 121	. . . . 5 ⊢ ((𝒫 1o = 2o ∧ 𝑥 ⊆ {∅}) → (𝑥 = ∅ ∨ 𝑥 = {∅}))
26	25	ex 114	. . . 4 ⊢ (𝒫 1o = 2o → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
27	26	alrimiv 1862	. . 3 ⊢ (𝒫 1o = 2o → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
28	27, 1	sylibr 133	. 2 ⊢ (𝒫 1o = 2o → EXMID)
29	18, 28	impbii 125	1 ⊢ (EXMID ↔ 𝒫 1o = 2o)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698  ∀wal 1341   = wceq 1343   ∈ wcel 2136   ⊆ wss 3116  ∅c0 3409  𝒫 cpw 3559  {csn 3576  {cpr 3577  EXMIDwem 4173  1_oc1o 6377  2_oc2o 6378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-nul 4108

This theorem depends on definitions:  df-bi 116  df-dc 825  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-exmid 4174  df-suc 4349  df-1o 6384  df-2o 6385

This theorem is referenced by:  pw1fin  6876  pw1nel3  7187  3nsssucpw1  7192  onntri35  7193