Theorem 2pwuninel 9158

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 9140	. . 3 ⊢ ¬ 𝒫 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴
2		elssuni 4898	. . . 4 ⊢ (𝒫 𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ⊆ ∪ 𝐴)
3		ssdomg 9034	. . . . 5 ⊢ (∪ 𝐴 ∈ V → (𝒫 𝒫 ∪ 𝐴 ⊆ ∪ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ≼ ∪ 𝐴))
4		canth2g 9157	. . . . . 6 ⊢ (∪ 𝐴 ∈ V → ∪ 𝐴 ≺ 𝒫 ∪ 𝐴)
5		pwexb 7784	. . . . . . 7 ⊢ (∪ 𝐴 ∈ V ↔ 𝒫 ∪ 𝐴 ∈ V)
6		canth2g 9157	. . . . . . 7 ⊢ (𝒫 ∪ 𝐴 ∈ V → 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴)
7	5, 6	sylbi 220	. . . . . 6 ⊢ (∪ 𝐴 ∈ V → 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴)
8		sdomtr 9141	. . . . . 6 ⊢ ((∪ 𝐴 ≺ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴) → ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴)
9	4, 7, 8	syl2anc 596	. . . . 5 ⊢ (∪ 𝐴 ∈ V → ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴)
10		domsdomtr 9138	. . . . . 6 ⊢ ((𝒫 𝒫 ∪ 𝐴 ≼ ∪ 𝐴 ∧ ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴) → 𝒫 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴)
11	10	ex 418	. . . . 5 ⊢ (𝒫 𝒫 ∪ 𝐴 ≼ ∪ 𝐴 → (∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴))
12	3, 9, 11	syl6ci 72	. . . 4 ⊢ (∪ 𝐴 ∈ V → (𝒫 𝒫 ∪ 𝐴 ⊆ ∪ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴))
13	2, 12	syl5 35	. . 3 ⊢ (∪ 𝐴 ∈ V → (𝒫 𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴))
14	1, 13	mtoi 202	. 2 ⊢ (∪ 𝐴 ∈ V → ¬ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴)
15		elex 3471	. . . 4 ⊢ (𝒫 𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ∈ V)
16		pwexb 7784	. . . . 5 ⊢ (𝒫 ∪ 𝐴 ∈ V ↔ 𝒫 𝒫 ∪ 𝐴 ∈ V)
17	5, 16	bitri 278	. . . 4 ⊢ (∪ 𝐴 ∈ V ↔ 𝒫 𝒫 ∪ 𝐴 ∈ V)
18	15, 17	sylibr 237	. . 3 ⊢ (𝒫 𝒫 ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 ∈ V)
19	18	con3i 155	. 2 ⊢ (¬ ∪ 𝐴 ∈ V → ¬ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴)
20	14, 19	pm2.61i 184	1 ⊢ ¬ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3896  𝒫 cpw 4556  ∪ cuni 4866   class class class wbr 5102   ≼ cdom 8978   ≺ csdm 8979

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-er 8724  df-en 8981  df-dom 8982  df-sdom 8983

This theorem is used by:  mnfnre  11359