Theorem exmidpw 6886

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 6408	. . . . 5 ⊢ 1o = {∅}
2		p0ex 4174	. . . . 5 ⊢ {∅} ∈ V
3	1, 2	eqeltri 2243	. . . 4 ⊢ 1o ∈ V
4	3	pwex 4169	. . 3 ⊢ 𝒫 1o ∈ V
5		exmid01 4184	. . . . . . . . 9 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
6	5	biimpi 119	. . . . . . . 8 ⊢ (EXMID → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
7	6	19.21bi 1551	. . . . . . 7 ⊢ (EXMID → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
8	1	pweqi 3570	. . . . . . . . 9 ⊢ 𝒫 1o = 𝒫 {∅}
9	8	eleq2i 2237	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 1o ↔ 𝑥 ∈ 𝒫 {∅})
10		velpw 3573	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 {∅} ↔ 𝑥 ⊆ {∅})
11	9, 10	bitri 183	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 1o ↔ 𝑥 ⊆ {∅})
12		vex 2733	. . . . . . . 8 ⊢ 𝑥 ∈ V
13	12	elpr 3604	. . . . . . 7 ⊢ (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
14	7, 11, 13	3imtr4g 204	. . . . . 6 ⊢ (EXMID → (𝑥 ∈ 𝒫 1o → 𝑥 ∈ {∅, {∅}}))
15	14	ssrdv 3153	. . . . 5 ⊢ (EXMID → 𝒫 1o ⊆ {∅, {∅}})
16		pwpw0ss 3791	. . . . . . 7 ⊢ {∅, {∅}} ⊆ 𝒫 {∅}
17	16, 8	sseqtrri 3182	. . . . . 6 ⊢ {∅, {∅}} ⊆ 𝒫 1o
18	17	a1i 9	. . . . 5 ⊢ (EXMID → {∅, {∅}} ⊆ 𝒫 1o)
19	15, 18	eqssd 3164	. . . 4 ⊢ (EXMID → 𝒫 1o = {∅, {∅}})
20		df2o2 6410	. . . 4 ⊢ 2o = {∅, {∅}}
21	19, 20	eqtr4di 2221	. . 3 ⊢ (EXMID → 𝒫 1o = 2o)
22		eqeng 6744	. . 3 ⊢ (𝒫 1o ∈ V → (𝒫 1o = 2o → 𝒫 1o ≈ 2o))
23	4, 21, 22	mpsyl 65	. 2 ⊢ (EXMID → 𝒫 1o ≈ 2o)
24		0nep0 4151	. . . . . . . 8 ⊢ ∅ ≠ {∅}
25		0ex 4116	. . . . . . . . . . 11 ⊢ ∅ ∈ V
26	25, 2	prss 3736	. . . . . . . . . 10 ⊢ ((∅ ∈ 𝒫 1o ∧ {∅} ∈ 𝒫 1o) ↔ {∅, {∅}} ⊆ 𝒫 1o)
27	17, 26	mpbir 145	. . . . . . . . 9 ⊢ (∅ ∈ 𝒫 1o ∧ {∅} ∈ 𝒫 1o)
28		en2eqpr 6885	. . . . . . . . . 10 ⊢ ((𝒫 1o ≈ 2o ∧ ∅ ∈ 𝒫 1o ∧ {∅} ∈ 𝒫 1o) → (∅ ≠ {∅} → 𝒫 1o = {∅, {∅}}))
29	28	3expb 1199	. . . . . . . . 9 ⊢ ((𝒫 1o ≈ 2o ∧ (∅ ∈ 𝒫 1o ∧ {∅} ∈ 𝒫 1o)) → (∅ ≠ {∅} → 𝒫 1o = {∅, {∅}}))
30	27, 29	mpan2 423	. . . . . . . 8 ⊢ (𝒫 1o ≈ 2o → (∅ ≠ {∅} → 𝒫 1o = {∅, {∅}}))
31	24, 30	mpi 15	. . . . . . 7 ⊢ (𝒫 1o ≈ 2o → 𝒫 1o = {∅, {∅}})
32	31	eleq2d 2240	. . . . . 6 ⊢ (𝒫 1o ≈ 2o → (𝑥 ∈ 𝒫 1o ↔ 𝑥 ∈ {∅, {∅}}))
33	32, 11, 13	3bitr3g 221	. . . . 5 ⊢ (𝒫 1o ≈ 2o → (𝑥 ⊆ {∅} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
34	33	biimpd 143	. . . 4 ⊢ (𝒫 1o ≈ 2o → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
35	34	alrimiv 1867	. . 3 ⊢ (𝒫 1o ≈ 2o → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
36	35, 5	sylibr 133	. 2 ⊢ (𝒫 1o ≈ 2o → EXMID)
37	23, 36	impbii 125	1 ⊢ (EXMID ↔ 𝒫 1o ≈ 2o)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 703  ∀wal 1346   = wceq 1348   ∈ wcel 2141   ≠ wne 2340  Vcvv 2730   ⊆ wss 3121  ∅c0 3414  𝒫 cpw 3566  {csn 3583  {cpr 3584   class class class wbr 3989  EXMIDwem 4180  1_oc1o 6388  2_oc2o 6389   ≈ cen 6716

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-exmid 4181  df-id 4278  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1o 6395  df-2o 6396  df-en 6719

This theorem is referenced by:  pwf1oexmid  14032