Theorem exmidpw 6964

Description: Excluded middle is equivalent to the power set of 1_o having two elements. Remark of [PradicBrown2022], 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 6482	. . . . 5 ⊢ 1o = {∅}
2		p0ex 4217	. . . . 5 ⊢ {∅} ∈ V
3	1, 2	eqeltri 2266	. . . 4 ⊢ 1o ∈ V
4	3	pwex 4212	. . 3 ⊢ 𝒫 1o ∈ V
5		exmid01 4227	. . . . . . . . 9 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
6	5	biimpi 120	. . . . . . . 8 ⊢ (EXMID → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
7	6	19.21bi 1569	. . . . . . 7 ⊢ (EXMID → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
8	1	pweqi 3605	. . . . . . . . 9 ⊢ 𝒫 1o = 𝒫 {∅}
9	8	eleq2i 2260	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 1o ↔ 𝑥 ∈ 𝒫 {∅})
10		velpw 3608	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 {∅} ↔ 𝑥 ⊆ {∅})
11	9, 10	bitri 184	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 1o ↔ 𝑥 ⊆ {∅})
12		vex 2763	. . . . . . . 8 ⊢ 𝑥 ∈ V
13	12	elpr 3639	. . . . . . 7 ⊢ (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
14	7, 11, 13	3imtr4g 205	. . . . . 6 ⊢ (EXMID → (𝑥 ∈ 𝒫 1o → 𝑥 ∈ {∅, {∅}}))
15	14	ssrdv 3185	. . . . 5 ⊢ (EXMID → 𝒫 1o ⊆ {∅, {∅}})
16		pwpw0ss 3830	. . . . . . 7 ⊢ {∅, {∅}} ⊆ 𝒫 {∅}
17	16, 8	sseqtrri 3214	. . . . . 6 ⊢ {∅, {∅}} ⊆ 𝒫 1o
18	17	a1i 9	. . . . 5 ⊢ (EXMID → {∅, {∅}} ⊆ 𝒫 1o)
19	15, 18	eqssd 3196	. . . 4 ⊢ (EXMID → 𝒫 1o = {∅, {∅}})
20		df2o2 6484	. . . 4 ⊢ 2o = {∅, {∅}}
21	19, 20	eqtr4di 2244	. . 3 ⊢ (EXMID → 𝒫 1o = 2o)
22		eqeng 6820	. . 3 ⊢ (𝒫 1o ∈ V → (𝒫 1o = 2o → 𝒫 1o ≈ 2o))
23	4, 21, 22	mpsyl 65	. 2 ⊢ (EXMID → 𝒫 1o ≈ 2o)
24		0nep0 4194	. . . . . . . 8 ⊢ ∅ ≠ {∅}
25		0ex 4156	. . . . . . . . . . 11 ⊢ ∅ ∈ V
26	25, 2	prss 3774	. . . . . . . . . 10 ⊢ ((∅ ∈ 𝒫 1o ∧ {∅} ∈ 𝒫 1o) ↔ {∅, {∅}} ⊆ 𝒫 1o)
27	17, 26	mpbir 146	. . . . . . . . 9 ⊢ (∅ ∈ 𝒫 1o ∧ {∅} ∈ 𝒫 1o)
28		en2eqpr 6963	. . . . . . . . . 10 ⊢ ((𝒫 1o ≈ 2o ∧ ∅ ∈ 𝒫 1o ∧ {∅} ∈ 𝒫 1o) → (∅ ≠ {∅} → 𝒫 1o = {∅, {∅}}))
29	28	3expb 1206	. . . . . . . . 9 ⊢ ((𝒫 1o ≈ 2o ∧ (∅ ∈ 𝒫 1o ∧ {∅} ∈ 𝒫 1o)) → (∅ ≠ {∅} → 𝒫 1o = {∅, {∅}}))
30	27, 29	mpan2 425	. . . . . . . 8 ⊢ (𝒫 1o ≈ 2o → (∅ ≠ {∅} → 𝒫 1o = {∅, {∅}}))
31	24, 30	mpi 15	. . . . . . 7 ⊢ (𝒫 1o ≈ 2o → 𝒫 1o = {∅, {∅}})
32	31	eleq2d 2263	. . . . . 6 ⊢ (𝒫 1o ≈ 2o → (𝑥 ∈ 𝒫 1o ↔ 𝑥 ∈ {∅, {∅}}))
33	32, 11, 13	3bitr3g 222	. . . . 5 ⊢ (𝒫 1o ≈ 2o → (𝑥 ⊆ {∅} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
34	33	biimpd 144	. . . 4 ⊢ (𝒫 1o ≈ 2o → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
35	34	alrimiv 1885	. . 3 ⊢ (𝒫 1o ≈ 2o → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
36	35, 5	sylibr 134	. 2 ⊢ (𝒫 1o ≈ 2o → EXMID)
37	23, 36	impbii 126	1 ⊢ (EXMID ↔ 𝒫 1o ≈ 2o)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  ∀wal 1362   = wceq 1364   ∈ wcel 2164   ≠ wne 2364  Vcvv 2760   ⊆ wss 3153  ∅c0 3446  𝒫 cpw 3601  {csn 3618  {cpr 3619   class class class wbr 4029  EXMIDwem 4223  1_oc1o 6462  2_oc2o 6463   ≈ cen 6792

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-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  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-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-exmid 4224  df-id 4324  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-1o 6469  df-2o 6470  df-en 6795

This theorem is referenced by:  exmidpw2en  6968  pwf1oexmid  15490