Theorem pwnss 4222

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 2274	. . . . . . 7 ⊢ ((𝑦 = {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∧ 𝑦 = {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥}) → (𝑦 ∈ 𝑦 ↔ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥}))
2	1	anidms 397	. . . . . 6 ⊢ (𝑦 = {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} → (𝑦 ∈ 𝑦 ↔ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥}))
3	2	notbid 671	. . . . 5 ⊢ (𝑦 = {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} → (¬ 𝑦 ∈ 𝑦 ↔ ¬ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥}))
4		df-nel 2476	. . . . . . 7 ⊢ (𝑥 ∉ 𝑥 ↔ ¬ 𝑥 ∈ 𝑥)
5		eleq12 2274	. . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
6	5	anidms 397	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
7	6	notbid 671	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
8	4, 7	bitrid 192	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∉ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
9	8	cbvrabv 2778	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} = {𝑦 ∈ 𝐴 ∣ ¬ 𝑦 ∈ 𝑦}
10	3, 9	elrab2 2942	. . . 4 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ↔ ({𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ 𝐴 ∧ ¬ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥}))
11		pclem6 1396	. . . 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 3198	. . 3 ⊢ (𝒫 𝐴 ⊆ 𝐴 → ({𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ 𝒫 𝐴 → {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ 𝐴))
14	12, 13	mtoi 668	. 2 ⊢ (𝒫 𝐴 ⊆ 𝐴 → ¬ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ 𝒫 𝐴)
15		ssrab2 3289	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ⊆ 𝐴
16		elpw2g 4219	. . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ 𝒫 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ⊆ 𝐴))
17	15, 16	mpbiri 168	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑥} ∈ 𝒫 𝐴)
18	14, 17	nsyl3 629	1 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1375   ∈ wcel 2180   ∉ wnel 2475  {crab 2492   ⊆ wss 3177  𝒫 cpw 3629

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191  ax-sep 4181

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-nel 2476  df-rab 2497  df-v 2781  df-in 3183  df-ss 3190  df-pw 3631

This theorem is referenced by:  pwne  4223  pwuninel2  6398