Theorem 2pwuninel 9060

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 9042	. . 3 ⊢ ¬ 𝒫 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴
2		elssuni 4869	. . . 4 ⊢ (𝒫 𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ⊆ ∪ 𝐴)
3		ssdomg 8937	. . . . 5 ⊢ (∪ 𝐴 ∈ V → (𝒫 𝒫 ∪ 𝐴 ⊆ ∪ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ≼ ∪ 𝐴))
4		canth2g 9059	. . . . . 6 ⊢ (∪ 𝐴 ∈ V → ∪ 𝐴 ≺ 𝒫 ∪ 𝐴)
5		pwexb 7709	. . . . . . 7 ⊢ (∪ 𝐴 ∈ V ↔ 𝒫 ∪ 𝐴 ∈ V)
6		canth2g 9059	. . . . . . 7 ⊢ (𝒫 ∪ 𝐴 ∈ V → 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴)
7	5, 6	sylbi 218	. . . . . 6 ⊢ (∪ 𝐴 ∈ V → 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴)
8		sdomtr 9043	. . . . . 6 ⊢ ((∪ 𝐴 ≺ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴) → ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴)
9	4, 7, 8	syl2anc 590	. . . . 5 ⊢ (∪ 𝐴 ∈ V → ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴)
10		domsdomtr 9040	. . . . . 6 ⊢ ((𝒫 𝒫 ∪ 𝐴 ≼ ∪ 𝐴 ∧ ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴) → 𝒫 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴)
11	10	ex 413	. . . . 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 200	. 2 ⊢ (∪ 𝐴 ∈ V → ¬ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴)
15		elex 3452	. . . 4 ⊢ (𝒫 𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ∈ V)
16		pwexb 7709	. . . . 5 ⊢ (𝒫 ∪ 𝐴 ∈ V ↔ 𝒫 𝒫 ∪ 𝐴 ∈ V)
17	5, 16	bitri 276	. . . 4 ⊢ (∪ 𝐴 ∈ V ↔ 𝒫 𝒫 ∪ 𝐴 ∈ V)
18	15, 17	sylibr 235	. . 3 ⊢ (𝒫 𝒫 ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 ∈ V)
19	18	con3i 154	. 2 ⊢ (¬ ∪ 𝐴 ∈ V → ¬ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴)
20	14, 19	pm2.61i 183	1 ⊢ ¬ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴

Syntax hints:  ¬ wn 3   ∈ wcel 2119  Vcvv 3431   ⊆ wss 3883  𝒫 cpw 4529  ∪ cuni 4838   class class class wbr 5072   ≼ cdom 8881   ≺ csdm 8882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886

This theorem is referenced by:  mnfnre  11179