Theorem 2pwuninel 9073

Description: The power set of the power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by NM, 27-Jun-2008.)

Assertion

Ref	Expression
2pwuninel	⊢ ¬ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴

Proof of Theorem 2pwuninel

Step	Hyp	Ref	Expression
1		sdomirr 9055	. . 3 ⊢ ¬ 𝒫 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴
2		elssuni 4897	. . . 4 ⊢ (𝒫 𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ⊆ ∪ 𝐴)
3		ssdomg 8948	. . . . 5 ⊢ (∪ 𝐴 ∈ V → (𝒫 𝒫 ∪ 𝐴 ⊆ ∪ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ≼ ∪ 𝐴))
4		canth2g 9072	. . . . . 6 ⊢ (∪ 𝐴 ∈ V → ∪ 𝐴 ≺ 𝒫 ∪ 𝐴)
5		pwexb 7722	. . . . . . 7 ⊢ (∪ 𝐴 ∈ V ↔ 𝒫 ∪ 𝐴 ∈ V)
6		canth2g 9072	. . . . . . 7 ⊢ (𝒫 ∪ 𝐴 ∈ V → 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴)
7	5, 6	sylbi 217	. . . . . 6 ⊢ (∪ 𝐴 ∈ V → 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴)
8		sdomtr 9056	. . . . . 6 ⊢ ((∪ 𝐴 ≺ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴) → ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴)
9	4, 7, 8	syl2anc 584	. . . . 5 ⊢ (∪ 𝐴 ∈ V → ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴)
10		domsdomtr 9053	. . . . . 6 ⊢ ((𝒫 𝒫 ∪ 𝐴 ≼ ∪ 𝐴 ∧ ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴) → 𝒫 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴)
11	10	ex 412	. . . . 5 ⊢ (𝒫 𝒫 ∪ 𝐴 ≼ ∪ 𝐴 → (∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴))
12	3, 9, 11	syl6ci 71	. . . 4 ⊢ (∪ 𝐴 ∈ V → (𝒫 𝒫 ∪ 𝐴 ⊆ ∪ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴))
13	2, 12	syl5 34	. . 3 ⊢ (∪ 𝐴 ∈ V → (𝒫 𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴))
14	1, 13	mtoi 199	. 2 ⊢ (∪ 𝐴 ∈ V → ¬ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴)
15		elex 3465	. . . 4 ⊢ (𝒫 𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ∈ V)
16		pwexb 7722	. . . . 5 ⊢ (𝒫 ∪ 𝐴 ∈ V ↔ 𝒫 𝒫 ∪ 𝐴 ∈ V)
17	5, 16	bitri 275	. . . 4 ⊢ (∪ 𝐴 ∈ V ↔ 𝒫 𝒫 ∪ 𝐴 ∈ V)
18	15, 17	sylibr 234	. . 3 ⊢ (𝒫 𝒫 ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 ∈ V)
19	18	con3i 154	. 2 ⊢ (¬ ∪ 𝐴 ∈ V → ¬ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴)
20	14, 19	pm2.61i 182	1 ⊢ ¬ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴

Syntax hints:  ¬ wn 3   ∈ wcel 2109  Vcvv 3444   ⊆ wss 3911  𝒫 cpw 4559  ∪ cuni 4867   class class class wbr 5102   ≼ cdom 8893   ≺ csdm 8894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898

This theorem is referenced by:  mnfnre  11193