Theorem bnj1533 30922

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1533.1	⊢ (𝜃 → ∀𝑧 ∈ 𝐵 ¬ 𝑧 ∈ 𝐷)
bnj1533.2	⊢ 𝐵 ⊆ 𝐴
bnj1533.3	⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ 𝐶 ≠ 𝐸}

Assertion

Ref	Expression
bnj1533	⊢ (𝜃 → ∀𝑧 ∈ 𝐵 𝐶 = 𝐸)

Proof of Theorem bnj1533

Step	Hyp	Ref	Expression
1		bnj1533.1	. . . 4 ⊢ (𝜃 → ∀𝑧 ∈ 𝐵 ¬ 𝑧 ∈ 𝐷)
2	1	bnj1211 30868	. . 3 ⊢ (𝜃 → ∀𝑧(𝑧 ∈ 𝐵 → ¬ 𝑧 ∈ 𝐷))
3		bnj1533.3	. . . . . . . 8 ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ 𝐶 ≠ 𝐸}
4	3	rabeq2i 3197	. . . . . . 7 ⊢ (𝑧 ∈ 𝐷 ↔ (𝑧 ∈ 𝐴 ∧ 𝐶 ≠ 𝐸))
5	4	notbii 310	. . . . . 6 ⊢ (¬ 𝑧 ∈ 𝐷 ↔ ¬ (𝑧 ∈ 𝐴 ∧ 𝐶 ≠ 𝐸))
6		imnan 438	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 → ¬ 𝐶 ≠ 𝐸) ↔ ¬ (𝑧 ∈ 𝐴 ∧ 𝐶 ≠ 𝐸))
7		nne 2798	. . . . . . 7 ⊢ (¬ 𝐶 ≠ 𝐸 ↔ 𝐶 = 𝐸)
8	7	imbi2i 326	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 → ¬ 𝐶 ≠ 𝐸) ↔ (𝑧 ∈ 𝐴 → 𝐶 = 𝐸))
9	5, 6, 8	3bitr2i 288	. . . . 5 ⊢ (¬ 𝑧 ∈ 𝐷 ↔ (𝑧 ∈ 𝐴 → 𝐶 = 𝐸))
10	9	imbi2i 326	. . . 4 ⊢ ((𝑧 ∈ 𝐵 → ¬ 𝑧 ∈ 𝐷) ↔ (𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)))
11		bnj1533.2	. . . . . . 7 ⊢ 𝐵 ⊆ 𝐴
12	11	sseli 3599	. . . . . 6 ⊢ (𝑧 ∈ 𝐵 → 𝑧 ∈ 𝐴)
13	12	imim1i 63	. . . . 5 ⊢ ((𝑧 ∈ 𝐴 → 𝐶 = 𝐸) → (𝑧 ∈ 𝐵 → 𝐶 = 𝐸))
14		ax-1 6	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 → 𝐶 = 𝐸) → (𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)))
15	14	anim1i 592	. . . . . . . 8 ⊢ (((𝑧 ∈ 𝐴 → 𝐶 = 𝐸) ∧ 𝑧 ∈ 𝐵) → ((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) ∧ 𝑧 ∈ 𝐵))
16		simpr 477	. . . . . . . . . 10 ⊢ (((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
17		simpl 473	. . . . . . . . . 10 ⊢ (((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) ∧ 𝑧 ∈ 𝐵) → (𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)))
18	16, 17	mpd 15	. . . . . . . . 9 ⊢ (((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) ∧ 𝑧 ∈ 𝐵) → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸))
19	18, 16	jca 554	. . . . . . . 8 ⊢ (((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) ∧ 𝑧 ∈ 𝐵) → ((𝑧 ∈ 𝐴 → 𝐶 = 𝐸) ∧ 𝑧 ∈ 𝐵))
20	15, 19	impbii 199	. . . . . . 7 ⊢ (((𝑧 ∈ 𝐴 → 𝐶 = 𝐸) ∧ 𝑧 ∈ 𝐵) ↔ ((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) ∧ 𝑧 ∈ 𝐵))
21	20	imbi1i 339	. . . . . 6 ⊢ ((((𝑧 ∈ 𝐴 → 𝐶 = 𝐸) ∧ 𝑧 ∈ 𝐵) → 𝐶 = 𝐸) ↔ (((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) ∧ 𝑧 ∈ 𝐵) → 𝐶 = 𝐸))
22		impexp 462	. . . . . 6 ⊢ ((((𝑧 ∈ 𝐴 → 𝐶 = 𝐸) ∧ 𝑧 ∈ 𝐵) → 𝐶 = 𝐸) ↔ ((𝑧 ∈ 𝐴 → 𝐶 = 𝐸) → (𝑧 ∈ 𝐵 → 𝐶 = 𝐸)))
23		impexp 462	. . . . . 6 ⊢ ((((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) ∧ 𝑧 ∈ 𝐵) → 𝐶 = 𝐸) ↔ ((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) → (𝑧 ∈ 𝐵 → 𝐶 = 𝐸)))
24	21, 22, 23	3bitr3i 290	. . . . 5 ⊢ (((𝑧 ∈ 𝐴 → 𝐶 = 𝐸) → (𝑧 ∈ 𝐵 → 𝐶 = 𝐸)) ↔ ((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) → (𝑧 ∈ 𝐵 → 𝐶 = 𝐸)))
25	13, 24	mpbi 220	. . . 4 ⊢ ((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) → (𝑧 ∈ 𝐵 → 𝐶 = 𝐸))
26	10, 25	sylbi 207	. . 3 ⊢ ((𝑧 ∈ 𝐵 → ¬ 𝑧 ∈ 𝐷) → (𝑧 ∈ 𝐵 → 𝐶 = 𝐸))
27	2, 26	sylg 1750	. 2 ⊢ (𝜃 → ∀𝑧(𝑧 ∈ 𝐵 → 𝐶 = 𝐸))
28	27	bnj1142 30860	1 ⊢ (𝜃 → ∀𝑧 ∈ 𝐵 𝐶 = 𝐸)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  {crab 2916   ⊆ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ne 2795  df-ral 2917  df-rab 2921  df-in 3581  df-ss 3588

This theorem is referenced by:  bnj1523  31139