Theorem pwnss 4091

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 2205	. . . . . . 7 ⊢ ((𝑦 = {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∧ 𝑦 = {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥}) → (𝑦 ∈ 𝑦 ↔ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥}))
2	1	anidms 395	. . . . . 6 ⊢ (𝑦 = {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} → (𝑦 ∈ 𝑦 ↔ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥}))
3	2	notbid 657	. . . . 5 ⊢ (𝑦 = {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} → (¬ 𝑦 ∈ 𝑦 ↔ ¬ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥}))
4		df-nel 2405	. . . . . . 7 ⊢ (𝑥 ∉ 𝑥 ↔ ¬ 𝑥 ∈ 𝑥)
5		eleq12 2205	. . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
6	5	anidms 395	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
7	6	notbid 657	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
8	4, 7	syl5bb 191	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∉ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
9	8	cbvrabv 2688	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} = {𝑦 ∈ 𝐴 ∣ ¬ 𝑦 ∈ 𝑦}
10	3, 9	elrab2 2847	. . . 4 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ↔ ({𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ 𝐴 ∧ ¬ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥}))
11		pclem6 1353	. . . 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 5	. . 3 ⊢ ¬ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ 𝐴
13		ssel 3096	. . 3 ⊢ (𝒫 𝐴 ⊆ 𝐴 → ({𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ 𝒫 𝐴 → {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ 𝐴))
14	12, 13	mtoi 654	. 2 ⊢ (𝒫 𝐴 ⊆ 𝐴 → ¬ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ 𝒫 𝐴)
15		ssrab2 3187	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ⊆ 𝐴
16		elpw2g 4089	. . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ 𝒫 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ⊆ 𝐴))
17	15, 16	mpbiri 167	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ 𝒫 𝐴)
18	14, 17	nsyl3 616	1 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481   ∉ wnel 2404  {crab 2421   ⊆ wss 3076  𝒫 cpw 3515

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-nel 2405  df-rab 2426  df-v 2691  df-in 3082  df-ss 3089  df-pw 3517

This theorem is referenced by:  pwne  4092  pwuninel2  6187