Theorem bnj1533 32124

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 32069	. . 3 ⊢ (𝜃 → ∀𝑧(𝑧 ∈ 𝐵 → ¬ 𝑧 ∈ 𝐷))
3		bnj1533.3	. . . . . . . 8 ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ 𝐶 ≠ 𝐸}
4	3	rabeq2i 3487	. . . . . . 7 ⊢ (𝑧 ∈ 𝐷 ↔ (𝑧 ∈ 𝐴 ∧ 𝐶 ≠ 𝐸))
5	4	notbii 322	. . . . . 6 ⊢ (¬ 𝑧 ∈ 𝐷 ↔ ¬ (𝑧 ∈ 𝐴 ∧ 𝐶 ≠ 𝐸))
6		imnan 402	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 → ¬ 𝐶 ≠ 𝐸) ↔ ¬ (𝑧 ∈ 𝐴 ∧ 𝐶 ≠ 𝐸))
7		nne 3020	. . . . . . 7 ⊢ (¬ 𝐶 ≠ 𝐸 ↔ 𝐶 = 𝐸)
8	7	imbi2i 338	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 → ¬ 𝐶 ≠ 𝐸) ↔ (𝑧 ∈ 𝐴 → 𝐶 = 𝐸))
9	5, 6, 8	3bitr2i 301	. . . . 5 ⊢ (¬ 𝑧 ∈ 𝐷 ↔ (𝑧 ∈ 𝐴 → 𝐶 = 𝐸))
10	9	imbi2i 338	. . . 4 ⊢ ((𝑧 ∈ 𝐵 → ¬ 𝑧 ∈ 𝐷) ↔ (𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)))
11		bnj1533.2	. . . . . . 7 ⊢ 𝐵 ⊆ 𝐴
12	11	sseli 3963	. . . . . 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 616	. . . . . . . 8 ⊢ (((𝑧 ∈ 𝐴 → 𝐶 = 𝐸) ∧ 𝑧 ∈ 𝐵) → ((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) ∧ 𝑧 ∈ 𝐵))
16		simpr 487	. . . . . . . . . 10 ⊢ (((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
17		simpl 485	. . . . . . . . . 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 514	. . . . . . . 8 ⊢ (((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) ∧ 𝑧 ∈ 𝐵) → ((𝑧 ∈ 𝐴 → 𝐶 = 𝐸) ∧ 𝑧 ∈ 𝐵))
20	15, 19	impbii 211	. . . . . . 7 ⊢ (((𝑧 ∈ 𝐴 → 𝐶 = 𝐸) ∧ 𝑧 ∈ 𝐵) ↔ ((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) ∧ 𝑧 ∈ 𝐵))
21	20	imbi1i 352	. . . . . 6 ⊢ ((((𝑧 ∈ 𝐴 → 𝐶 = 𝐸) ∧ 𝑧 ∈ 𝐵) → 𝐶 = 𝐸) ↔ (((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) ∧ 𝑧 ∈ 𝐵) → 𝐶 = 𝐸))
22		impexp 453	. . . . . 6 ⊢ ((((𝑧 ∈ 𝐴 → 𝐶 = 𝐸) ∧ 𝑧 ∈ 𝐵) → 𝐶 = 𝐸) ↔ ((𝑧 ∈ 𝐴 → 𝐶 = 𝐸) → (𝑧 ∈ 𝐵 → 𝐶 = 𝐸)))
23		impexp 453	. . . . . 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 303	. . . . 5 ⊢ (((𝑧 ∈ 𝐴 → 𝐶 = 𝐸) → (𝑧 ∈ 𝐵 → 𝐶 = 𝐸)) ↔ ((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) → (𝑧 ∈ 𝐵 → 𝐶 = 𝐸)))
25	13, 24	mpbi 232	. . . 4 ⊢ ((𝑧 ∈ 𝐵 → (𝑧 ∈ 𝐴 → 𝐶 = 𝐸)) → (𝑧 ∈ 𝐵 → 𝐶 = 𝐸))
26	10, 25	sylbi 219	. . 3 ⊢ ((𝑧 ∈ 𝐵 → ¬ 𝑧 ∈ 𝐷) → (𝑧 ∈ 𝐵 → 𝐶 = 𝐸))
27	2, 26	sylg 1823	. 2 ⊢ (𝜃 → ∀𝑧(𝑧 ∈ 𝐵 → 𝐶 = 𝐸))
28	27	bnj1142 32061	1 ⊢ (𝜃 → ∀𝑧 ∈ 𝐵 𝐶 = 𝐸)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  {crab 3142   ⊆ wss 3936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-ne 3017  df-ral 3143  df-rab 3147  df-in 3943  df-ss 3952

This theorem is referenced by:  bnj1523  32343