Theorem 2pwuninel 9045

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 9027	. . 3 ⊢ ¬ 𝒫 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴
2		elssuni 4887	. . . 4 ⊢ (𝒫 𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ⊆ ∪ 𝐴)
3		ssdomg 8922	. . . . 5 ⊢ (∪ 𝐴 ∈ V → (𝒫 𝒫 ∪ 𝐴 ⊆ ∪ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ≼ ∪ 𝐴))
4		canth2g 9044	. . . . . 6 ⊢ (∪ 𝐴 ∈ V → ∪ 𝐴 ≺ 𝒫 ∪ 𝐴)
5		pwexb 7699	. . . . . . 7 ⊢ (∪ 𝐴 ∈ V ↔ 𝒫 ∪ 𝐴 ∈ V)
6		canth2g 9044	. . . . . . 7 ⊢ (𝒫 ∪ 𝐴 ∈ V → 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴)
7	5, 6	sylbi 217	. . . . . 6 ⊢ (∪ 𝐴 ∈ V → 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴)
8		sdomtr 9028	. . . . . 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 9025	. . . . . 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 3457	. . . 4 ⊢ (𝒫 𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ∈ V)
16		pwexb 7699	. . . . 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 2111  Vcvv 3436   ⊆ wss 3897  𝒫 cpw 4547  ∪ cuni 4856   class class class wbr 5089   ≼ cdom 8867   ≺ csdm 8868

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872

This theorem is referenced by:  mnfnre  11155