Theorem exmidpweq 7067

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 4281	. . . . . . . 8 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
2	1	biimpi 120	. . . . . . 7 ⊢ (EXMID → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
3	2	19.21bi 1604	. . . . . 6 ⊢ (EXMID → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
4		df1o2 6573	. . . . . . . . 9 ⊢ 1o = {∅}
5	4	pweqi 3653	. . . . . . . 8 ⊢ 𝒫 1o = 𝒫 {∅}
6	5	eleq2i 2296	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 1o ↔ 𝑥 ∈ 𝒫 {∅})
7		velpw 3656	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 {∅} ↔ 𝑥 ⊆ {∅})
8	6, 7	bitri 184	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 1o ↔ 𝑥 ⊆ {∅})
9		vex 2802	. . . . . . 7 ⊢ 𝑥 ∈ V
10	9	elpr 3687	. . . . . 6 ⊢ (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
11	3, 8, 10	3imtr4g 205	. . . . 5 ⊢ (EXMID → (𝑥 ∈ 𝒫 1o → 𝑥 ∈ {∅, {∅}}))
12	11	ssrdv 3230	. . . 4 ⊢ (EXMID → 𝒫 1o ⊆ {∅, {∅}})
13		pwpw0ss 3882	. . . . . 6 ⊢ {∅, {∅}} ⊆ 𝒫 {∅}
14	13, 5	sseqtrri 3259	. . . . 5 ⊢ {∅, {∅}} ⊆ 𝒫 1o
15	14	a1i 9	. . . 4 ⊢ (EXMID → {∅, {∅}} ⊆ 𝒫 1o)
16	12, 15	eqssd 3241	. . 3 ⊢ (EXMID → 𝒫 1o = {∅, {∅}})
17		df2o2 6575	. . 3 ⊢ 2o = {∅, {∅}}
18	16, 17	eqtr4di 2280	. 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 2323	. . . . . . 7 ⊢ ((𝒫 1o = 2o ∧ 𝑥 ⊆ {∅}) → 𝑥 ∈ 𝒫 1o)
22		simpl 109	. . . . . . . 8 ⊢ ((𝒫 1o = 2o ∧ 𝑥 ⊆ {∅}) → 𝒫 1o = 2o)
23	22, 17	eqtrdi 2278	. . . . . . 7 ⊢ ((𝒫 1o = 2o ∧ 𝑥 ⊆ {∅}) → 𝒫 1o = {∅, {∅}})
24	21, 23	eleqtrd 2308	. . . . . 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 1920	. . 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 713  ∀wal 1393   = wceq 1395   ∈ wcel 2200   ⊆ wss 3197  ∅c0 3491  𝒫 cpw 3649  {csn 3666  {cpr 3667  EXMIDwem 4277  1_oc1o 6553  2_oc2o 6554

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-nul 4209

This theorem depends on definitions:  df-bi 117  df-dc 840  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-exmid 4278  df-suc 4461  df-1o 6560  df-2o 6561

This theorem is referenced by:  pw1fin  7068  pw1nel3  7412  3nsssucpw1  7417  onntri35  7418