Theorem bnj1533 32732

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 32677	. . 3 ⊢ (𝜃 → ∀𝑧(𝑧 ∈ 𝐵 → ¬ 𝑧 ∈ 𝐷))
3		bnj1533.3	. . . . . . . 8 ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ 𝐶 ≠ 𝐸}
4	3	rabeq2i 3412	. . . . . . 7 ⊢ (𝑧 ∈ 𝐷 ↔ (𝑧 ∈ 𝐴 ∧ 𝐶 ≠ 𝐸))
5	4	notbii 319	. . . . . 6 ⊢ (¬ 𝑧 ∈ 𝐷 ↔ ¬ (𝑧 ∈ 𝐴 ∧ 𝐶 ≠ 𝐸))
6		imnan 399	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 → ¬ 𝐶 ≠ 𝐸) ↔ ¬ (𝑧 ∈ 𝐴 ∧ 𝐶 ≠ 𝐸))
7		nne 2946	. . . . . . 7 ⊢ (¬ 𝐶 ≠ 𝐸 ↔ 𝐶 = 𝐸)
8	7	imbi2i 335	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 → ¬ 𝐶 ≠ 𝐸) ↔ (𝑧 ∈ 𝐴 → 𝐶 = 𝐸))
9	5, 6, 8	3bitr2i 298	. . . . 5 ⊢ (¬ 𝑧 ∈ 𝐷 ↔ (𝑧 ∈ 𝐴 → 𝐶 = 𝐸))
10	9	imbi2i 335	. . . 4 ⊢ ((𝑧 ∈ 𝐵 → ¬ 𝑧 ∈ 𝐷) ↔ (𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)))
11		bnj1533.2	. . . . . . 7 ⊢ 𝐵 ⊆ 𝐴
12	11	sseli 3913	. . . . . 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 614	. . . . . . . 8 ⊢ (((𝑧 ∈ 𝐴 → 𝐶 = 𝐸) ∧ 𝑧 ∈ 𝐵) → ((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) ∧ 𝑧 ∈ 𝐵))
16		simpr 484	. . . . . . . . . 10 ⊢ (((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
17		simpl 482	. . . . . . . . . 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 511	. . . . . . . 8 ⊢ (((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) ∧ 𝑧 ∈ 𝐵) → ((𝑧 ∈ 𝐴 → 𝐶 = 𝐸) ∧ 𝑧 ∈ 𝐵))
20	15, 19	impbii 208	. . . . . . 7 ⊢ (((𝑧 ∈ 𝐴 → 𝐶 = 𝐸) ∧ 𝑧 ∈ 𝐵) ↔ ((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) ∧ 𝑧 ∈ 𝐵))
21	20	imbi1i 349	. . . . . 6 ⊢ ((((𝑧 ∈ 𝐴 → 𝐶 = 𝐸) ∧ 𝑧 ∈ 𝐵) → 𝐶 = 𝐸) ↔ (((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) ∧ 𝑧 ∈ 𝐵) → 𝐶 = 𝐸))
22		impexp 450	. . . . . 6 ⊢ ((((𝑧 ∈ 𝐴 → 𝐶 = 𝐸) ∧ 𝑧 ∈ 𝐵) → 𝐶 = 𝐸) ↔ ((𝑧 ∈ 𝐴 → 𝐶 = 𝐸) → (𝑧 ∈ 𝐵 → 𝐶 = 𝐸)))
23		impexp 450	. . . . . 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 300	. . . . 5 ⊢ (((𝑧 ∈ 𝐴 → 𝐶 = 𝐸) → (𝑧 ∈ 𝐵 → 𝐶 = 𝐸)) ↔ ((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) → (𝑧 ∈ 𝐵 → 𝐶 = 𝐸)))
25	13, 24	mpbi 229	. . . 4 ⊢ ((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) → (𝑧 ∈ 𝐵 → 𝐶 = 𝐸))
26	10, 25	sylbi 216	. . 3 ⊢ ((𝑧 ∈ 𝐵 → ¬ 𝑧 ∈ 𝐷) → (𝑧 ∈ 𝐵 → 𝐶 = 𝐸))
27	2, 26	sylg 1826	. 2 ⊢ (𝜃 → ∀𝑧(𝑧 ∈ 𝐵 → 𝐶 = 𝐸))
28	27	bnj1142 32669	1 ⊢ (𝜃 → ∀𝑧 ∈ 𝐵 𝐶 = 𝐸)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  {crab 3067   ⊆ wss 3883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rab 3072  df-v 3424  df-in 3890  df-ss 3900

This theorem is referenced by:  bnj1523  32951