Theorem eldifpw 7489

Description: Membership in a power class difference. (Contributed by NM, 25-Mar-2007.)

Hypothesis

Ref	Expression
eldifpw.1	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
eldifpw	⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶 ⊆ 𝐵) → (𝐴 ∪ 𝐶) ∈ (𝒫 (𝐵 ∪ 𝐶) ∖ 𝒫 𝐵))

Proof of Theorem eldifpw

Step	Hyp	Ref	Expression
1		elpwi 4547	. . . 4 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵)
2		unss1 4154	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶))
3		eldifpw.1	. . . . . . 7 ⊢ 𝐶 ∈ V
4		unexg 7471	. . . . . . 7 ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ 𝐶 ∈ V) → (𝐴 ∪ 𝐶) ∈ V)
5	3, 4	mpan2 689	. . . . . 6 ⊢ (𝐴 ∈ 𝒫 𝐵 → (𝐴 ∪ 𝐶) ∈ V)
6		elpwg 4541	. . . . . 6 ⊢ ((𝐴 ∪ 𝐶) ∈ V → ((𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶)))
7	5, 6	syl 17	. . . . 5 ⊢ (𝐴 ∈ 𝒫 𝐵 → ((𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶)))
8	2, 7	syl5ibr 248	. . . 4 ⊢ (𝐴 ∈ 𝒫 𝐵 → (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶)))
9	1, 8	mpd 15	. . 3 ⊢ (𝐴 ∈ 𝒫 𝐵 → (𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶))
10		elpwi 4547	. . . . 5 ⊢ ((𝐴 ∪ 𝐶) ∈ 𝒫 𝐵 → (𝐴 ∪ 𝐶) ⊆ 𝐵)
11	10	unssbd 4163	. . . 4 ⊢ ((𝐴 ∪ 𝐶) ∈ 𝒫 𝐵 → 𝐶 ⊆ 𝐵)
12	11	con3i 157	. . 3 ⊢ (¬ 𝐶 ⊆ 𝐵 → ¬ (𝐴 ∪ 𝐶) ∈ 𝒫 𝐵)
13	9, 12	anim12i 614	. 2 ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶 ⊆ 𝐵) → ((𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶) ∧ ¬ (𝐴 ∪ 𝐶) ∈ 𝒫 𝐵))
14		eldif 3945	. 2 ⊢ ((𝐴 ∪ 𝐶) ∈ (𝒫 (𝐵 ∪ 𝐶) ∖ 𝒫 𝐵) ↔ ((𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶) ∧ ¬ (𝐴 ∪ 𝐶) ∈ 𝒫 𝐵))
15	13, 14	sylibr 236	1 ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶 ⊆ 𝐵) → (𝐴 ∪ 𝐶) ∈ (𝒫 (𝐵 ∪ 𝐶) ∖ 𝒫 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∈ wcel 2110  Vcvv 3494   ∖ cdif 3932   ∪ cun 3933   ⊆ wss 3935  𝒫 cpw 4538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-pw 4540  df-sn 4567  df-pr 4569  df-uni 4838

This theorem is referenced by: (None)