Theorem pwnss 3933

Description: The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Assertion

Ref	Expression
pwnss	⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴)

Proof of Theorem pwnss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq12 2143	. . . . . . 7 ⊢ ((𝑦 = {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∧ 𝑦 = {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥}) → (𝑦 ∈ 𝑦 ↔ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥}))
2	1	anidms 389	. . . . . 6 ⊢ (𝑦 = {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} → (𝑦 ∈ 𝑦 ↔ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥}))
3	2	notbid 624	. . . . 5 ⊢ (𝑦 = {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} → (¬ 𝑦 ∈ 𝑦 ↔ ¬ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥}))
4		df-nel 2340	. . . . . . 7 ⊢ (𝑥 ∉ 𝑥 ↔ ¬ 𝑥 ∈ 𝑥)
5		eleq12 2143	. . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
6	5	anidms 389	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
7	6	notbid 624	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
8	4, 7	syl5bb 190	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∉ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
9	8	cbvrabv 2600	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} = {𝑦 ∈ 𝐴 ∣ ¬ 𝑦 ∈ 𝑦}
10	3, 9	elrab2 2751	. . . 4 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ↔ ({𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ 𝐴 ∧ ¬ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥}))
11		pclem6 1305	. . . 4 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ↔ ({𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ 𝐴 ∧ ¬ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥})) → ¬ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ 𝐴)
12	10, 11	ax-mp 7	. . 3 ⊢ ¬ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ 𝐴
13		ssel 2993	. . 3 ⊢ (𝒫 𝐴 ⊆ 𝐴 → ({𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ 𝒫 𝐴 → {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ 𝐴))
14	12, 13	mtoi 622	. 2 ⊢ (𝒫 𝐴 ⊆ 𝐴 → ¬ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ 𝒫 𝐴)
15		ssrab2 3079	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ⊆ 𝐴
16		elpw2g 3931	. . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ 𝒫 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ⊆ 𝐴))
17	15, 16	mpbiri 166	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ 𝒫 𝐴)
18	14, 17	nsyl3 588	1 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433   ∉ wnel 2339  {crab 2352   ⊆ wss 2973  𝒫 cpw 3382

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-nel 2340  df-rab 2357  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384

This theorem is referenced by:  pwne  3934  pwuninel2  5920