Theorem eldifpw 4471

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 3581	. . . 4 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵)
2		unss1 3302	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶))
3		eldifpw.1	. . . . . . 7 ⊢ 𝐶 ∈ V
4		unexg 4437	. . . . . . 7 ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ 𝐶 ∈ V) → (𝐴 ∪ 𝐶) ∈ V)
5	3, 4	mpan2 425	. . . . . 6 ⊢ (𝐴 ∈ 𝒫 𝐵 → (𝐴 ∪ 𝐶) ∈ V)
6		elpwg 3580	. . . . . 6 ⊢ ((𝐴 ∪ 𝐶) ∈ V → ((𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶)))
7	5, 6	syl 14	. . . . 5 ⊢ (𝐴 ∈ 𝒫 𝐵 → ((𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶)))
8	2, 7	syl5ibr 156	. . . 4 ⊢ (𝐴 ∈ 𝒫 𝐵 → (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶)))
9	1, 8	mpd 13	. . 3 ⊢ (𝐴 ∈ 𝒫 𝐵 → (𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶))
10		elpwi 3581	. . . . 5 ⊢ ((𝐴 ∪ 𝐶) ∈ 𝒫 𝐵 → (𝐴 ∪ 𝐶) ⊆ 𝐵)
11	10	unssbd 3311	. . . 4 ⊢ ((𝐴 ∪ 𝐶) ∈ 𝒫 𝐵 → 𝐶 ⊆ 𝐵)
12	11	con3i 632	. . 3 ⊢ (¬ 𝐶 ⊆ 𝐵 → ¬ (𝐴 ∪ 𝐶) ∈ 𝒫 𝐵)
13	9, 12	anim12i 338	. 2 ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶 ⊆ 𝐵) → ((𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶) ∧ ¬ (𝐴 ∪ 𝐶) ∈ 𝒫 𝐵))
14		eldif 3136	. 2 ⊢ ((𝐴 ∪ 𝐶) ∈ (𝒫 (𝐵 ∪ 𝐶) ∖ 𝒫 𝐵) ↔ ((𝐴 ∪ 𝐶) ∈ 𝒫 (𝐵 ∪ 𝐶) ∧ ¬ (𝐴 ∪ 𝐶) ∈ 𝒫 𝐵))
15	13, 14	sylibr 134	1 ⊢ ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶 ⊆ 𝐵) → (𝐴 ∪ 𝐶) ∈ (𝒫 (𝐵 ∪ 𝐶) ∖ 𝒫 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2146  Vcvv 2735   ∖ cdif 3124   ∪ cun 3125   ⊆ wss 3127  𝒫 cpw 3572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pr 4203  ax-un 4427

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-v 2737  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-uni 3806

This theorem is referenced by: (None)