Theorem exmidpw 6551

Description: Excluded middle is equivalent to the power set of 1_𝑜 having two elements. Remark of [PradicBrown2021], p. 2. (Contributed by Jim Kingdon, 30-Jun-2022.)

Assertion

Ref	Expression
exmidpw	⊢ (EXMID ↔ 𝒫 1𝑜 ≈ 2𝑜)

Proof of Theorem exmidpw

Step	Hyp	Ref	Expression
1		df1o2 6126	. . . . 5 ⊢ 1𝑜 = {∅}
2		p0ex 3987	. . . . 5 ⊢ {∅} ∈ V
3	1, 2	eqeltri 2155	. . . 4 ⊢ 1𝑜 ∈ V
4	3	pwex 3982	. . 3 ⊢ 𝒫 1𝑜 ∈ V
5		exmid01 3996	. . . . . . . . 9 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
6	5	biimpi 118	. . . . . . . 8 ⊢ (EXMID → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
7	6	19.21bi 1491	. . . . . . 7 ⊢ (EXMID → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
8	1	pweqi 3410	. . . . . . . . 9 ⊢ 𝒫 1𝑜 = 𝒫 {∅}
9	8	eleq2i 2149	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 1𝑜 ↔ 𝑥 ∈ 𝒫 {∅})
10		selpw 3413	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 {∅} ↔ 𝑥 ⊆ {∅})
11	9, 10	bitri 182	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 1𝑜 ↔ 𝑥 ⊆ {∅})
12		vex 2615	. . . . . . . 8 ⊢ 𝑥 ∈ V
13	12	elpr 3443	. . . . . . 7 ⊢ (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
14	7, 11, 13	3imtr4g 203	. . . . . 6 ⊢ (EXMID → (𝑥 ∈ 𝒫 1𝑜 → 𝑥 ∈ {∅, {∅}}))
15	14	ssrdv 3016	. . . . 5 ⊢ (EXMID → 𝒫 1𝑜 ⊆ {∅, {∅}})
16		pwpw0ss 3622	. . . . . . 7 ⊢ {∅, {∅}} ⊆ 𝒫 {∅}
17	16, 8	sseqtr4i 3043	. . . . . 6 ⊢ {∅, {∅}} ⊆ 𝒫 1𝑜
18	17	a1i 9	. . . . 5 ⊢ (EXMID → {∅, {∅}} ⊆ 𝒫 1𝑜)
19	15, 18	eqssd 3027	. . . 4 ⊢ (EXMID → 𝒫 1𝑜 = {∅, {∅}})
20		df2o2 6128	. . . 4 ⊢ 2𝑜 = {∅, {∅}}
21	19, 20	syl6eqr 2133	. . 3 ⊢ (EXMID → 𝒫 1𝑜 = 2𝑜)
22		eqeng 6413	. . 3 ⊢ (𝒫 1𝑜 ∈ V → (𝒫 1𝑜 = 2𝑜 → 𝒫 1𝑜 ≈ 2𝑜))
23	4, 21, 22	mpsyl 64	. 2 ⊢ (EXMID → 𝒫 1𝑜 ≈ 2𝑜)
24		0nep0 3965	. . . . . . . 8 ⊢ ∅ ≠ {∅}
25		0ex 3931	. . . . . . . . . . 11 ⊢ ∅ ∈ V
26	25, 2	prss 3567	. . . . . . . . . 10 ⊢ ((∅ ∈ 𝒫 1𝑜 ∧ {∅} ∈ 𝒫 1𝑜) ↔ {∅, {∅}} ⊆ 𝒫 1𝑜)
27	17, 26	mpbir 144	. . . . . . . . 9 ⊢ (∅ ∈ 𝒫 1𝑜 ∧ {∅} ∈ 𝒫 1𝑜)
28		en2eqpr 6550	. . . . . . . . . 10 ⊢ ((𝒫 1𝑜 ≈ 2𝑜 ∧ ∅ ∈ 𝒫 1𝑜 ∧ {∅} ∈ 𝒫 1𝑜) → (∅ ≠ {∅} → 𝒫 1𝑜 = {∅, {∅}}))
29	28	3expb 1140	. . . . . . . . 9 ⊢ ((𝒫 1𝑜 ≈ 2𝑜 ∧ (∅ ∈ 𝒫 1𝑜 ∧ {∅} ∈ 𝒫 1𝑜)) → (∅ ≠ {∅} → 𝒫 1𝑜 = {∅, {∅}}))
30	27, 29	mpan2 416	. . . . . . . 8 ⊢ (𝒫 1𝑜 ≈ 2𝑜 → (∅ ≠ {∅} → 𝒫 1𝑜 = {∅, {∅}}))
31	24, 30	mpi 15	. . . . . . 7 ⊢ (𝒫 1𝑜 ≈ 2𝑜 → 𝒫 1𝑜 = {∅, {∅}})
32	31	eleq2d 2152	. . . . . 6 ⊢ (𝒫 1𝑜 ≈ 2𝑜 → (𝑥 ∈ 𝒫 1𝑜 ↔ 𝑥 ∈ {∅, {∅}}))
33	32, 11, 13	3bitr3g 220	. . . . 5 ⊢ (𝒫 1𝑜 ≈ 2𝑜 → (𝑥 ⊆ {∅} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
34	33	biimpd 142	. . . 4 ⊢ (𝒫 1𝑜 ≈ 2𝑜 → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
35	34	alrimiv 1797	. . 3 ⊢ (𝒫 1𝑜 ≈ 2𝑜 → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
36	35, 5	sylibr 132	. 2 ⊢ (𝒫 1𝑜 ≈ 2𝑜 → EXMID)
37	23, 36	impbii 124	1 ⊢ (EXMID ↔ 𝒫 1𝑜 ≈ 2𝑜)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 662  ∀wal 1283   = wceq 1285   ∈ wcel 1434   ≠ wne 2249  Vcvv 2612   ⊆ wss 2984  ∅c0 3269  𝒫 cpw 3406  {csn 3422  {cpr 3423   class class class wbr 3811  EXMIDwem 3993  1_𝑜c1o 6106  2_𝑜c2o 6107   ≈ cen 6385

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-exmid 3994  df-id 4084  df-suc 4162  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-1o 6113  df-2o 6114  df-en 6388

This theorem is referenced by: (None)