Theorem exmidpw 7181

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 6674	. . . . 5 ⊢ 1o = {∅}
2		p0ex 4306	. . . . 5 ⊢ {∅} ∈ V
3	1, 2	eqeltri 2307	. . . 4 ⊢ 1o ∈ V
4	3	pwex 4301	. . 3 ⊢ 𝒫 1o ∈ V
5		exmid01 4316	. . . . . . . . 9 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
6	5	biimpi 120	. . . . . . . 8 ⊢ (EXMID → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
7	6	19.21bi 1607	. . . . . . 7 ⊢ (EXMID → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
8	1	pweqi 3678	. . . . . . . . 9 ⊢ 𝒫 1o = 𝒫 {∅}
9	8	eleq2i 2301	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 1o ↔ 𝑥 ∈ 𝒫 {∅})
10		velpw 3681	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 {∅} ↔ 𝑥 ⊆ {∅})
11	9, 10	bitri 184	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 1o ↔ 𝑥 ⊆ {∅})
12		vex 2818	. . . . . . . 8 ⊢ 𝑥 ∈ V
13	12	elpr 3715	. . . . . . 7 ⊢ (𝑥 ∈ {∅, {∅}} ↔ (𝑥 = ∅ ∨ 𝑥 = {∅}))
14	7, 11, 13	3imtr4g 205	. . . . . 6 ⊢ (EXMID → (𝑥 ∈ 𝒫 1o → 𝑥 ∈ {∅, {∅}}))
15	14	ssrdv 3248	. . . . 5 ⊢ (EXMID → 𝒫 1o ⊆ {∅, {∅}})
16		pwpw0ss 3914	. . . . . . 7 ⊢ {∅, {∅}} ⊆ 𝒫 {∅}
17	16, 8	sseqtrri 3277	. . . . . 6 ⊢ {∅, {∅}} ⊆ 𝒫 1o
18	17	a1i 9	. . . . 5 ⊢ (EXMID → {∅, {∅}} ⊆ 𝒫 1o)
19	15, 18	eqssd 3259	. . . 4 ⊢ (EXMID → 𝒫 1o = {∅, {∅}})
20		df2o2 6676	. . . 4 ⊢ 2o = {∅, {∅}}
21	19, 20	eqtr4di 2285	. . 3 ⊢ (EXMID → 𝒫 1o = 2o)
22		eqeng 7018	. . 3 ⊢ (𝒫 1o ∈ V → (𝒫 1o = 2o → 𝒫 1o ≈ 2o))
23	4, 21, 22	mpsyl 65	. 2 ⊢ (EXMID → 𝒫 1o ≈ 2o)
24		0nep0 4283	. . . . . . . 8 ⊢ ∅ ≠ {∅}
25		0ex 4242	. . . . . . . . . . 11 ⊢ ∅ ∈ V
26	25, 2	prss 3855	. . . . . . . . . 10 ⊢ ((∅ ∈ 𝒫 1o ∧ {∅} ∈ 𝒫 1o) ↔ {∅, {∅}} ⊆ 𝒫 1o)
27	17, 26	mpbir 146	. . . . . . . . 9 ⊢ (∅ ∈ 𝒫 1o ∧ {∅} ∈ 𝒫 1o)
28		en2eqpr 7180	. . . . . . . . . 10 ⊢ ((𝒫 1o ≈ 2o ∧ ∅ ∈ 𝒫 1o ∧ {∅} ∈ 𝒫 1o) → (∅ ≠ {∅} → 𝒫 1o = {∅, {∅}}))
29	28	3expb 1231	. . . . . . . . 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 2304	. . . . . 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 1923	. . 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 716  ∀wal 1396   = wceq 1398   ∈ wcel 2205   ≠ wne 2414  Vcvv 2815   ⊆ wss 3214  ∅c0 3512  𝒫 cpw 3674  {csn 3694  {cpr 3695   class class class wbr 4114  EXMIDwem 4312  1_oc1o 6653  2_oc2o 6654   ≈ cen 6986

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-exmid 4313  df-id 4419  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-1o 6660  df-2o 6661  df-en 6989

This theorem is referenced by:  exmidpw2en  7185  pwf1oexmid  16912