Theorem bnj1154 35188

Description: Property of Fr. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
bnj1154	⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ∧ 𝐵 ∈ V) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦

Proof of Theorem bnj1154

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj658 34941	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ∧ 𝐵 ∈ V) → (𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅))
2		elisset 2822	. . . . 5 ⊢ (𝐵 ∈ V → ∃𝑏 𝑏 = 𝐵)
3	2	bnj708 34946	. . . 4 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ∧ 𝐵 ∈ V) → ∃𝑏 𝑏 = 𝐵)
4		df-fr 5578	. . . . . . . 8 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑏((𝑏 ⊆ 𝐴 ∧ 𝑏 ≠ ∅) → ∃𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ¬ 𝑦𝑅𝑥))
5	4	biimpi 217	. . . . . . 7 ⊢ (𝑅 Fr 𝐴 → ∀𝑏((𝑏 ⊆ 𝐴 ∧ 𝑏 ≠ ∅) → ∃𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ¬ 𝑦𝑅𝑥))
6	5	19.21bi 2201	. . . . . 6 ⊢ (𝑅 Fr 𝐴 → ((𝑏 ⊆ 𝐴 ∧ 𝑏 ≠ ∅) → ∃𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ¬ 𝑦𝑅𝑥))
7	6	3impib 1122	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑏 ⊆ 𝐴 ∧ 𝑏 ≠ ∅) → ∃𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ¬ 𝑦𝑅𝑥)
8		sseq1 3947	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
9		neeq1 2997	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏 ≠ ∅ ↔ 𝐵 ≠ ∅))
10	8, 9	3anbi23d 1447	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝑅 Fr 𝐴 ∧ 𝑏 ⊆ 𝐴 ∧ 𝑏 ≠ ∅) ↔ (𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)))
11		raleq 3295	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
12	11	rexeqbi1dv 3309	. . . . . 6 ⊢ (𝑏 = 𝐵 → (∃𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
13	10, 12	imbi12d 345	. . . . 5 ⊢ (𝑏 = 𝐵 → (((𝑅 Fr 𝐴 ∧ 𝑏 ⊆ 𝐴 ∧ 𝑏 ≠ ∅) → ∃𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ¬ 𝑦𝑅𝑥) ↔ ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)))
14	7, 13	mpbii 234	. . . 4 ⊢ (𝑏 = 𝐵 → ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
15	3, 14	bnj593 34935	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ∧ 𝐵 ∈ V) → ∃𝑏((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
16	15	bnj937 34961	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ∧ 𝐵 ∈ V) → ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
17	1, 16	mpd 15	1 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ∧ 𝐵 ∈ V) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1092  ∀wal 1545   = wceq 1547  ∃wex 1786   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054  ∃wrex 3064  Vcvv 3432   ⊆ wss 3890  ∅c0 4268   class class class wbr 5079   Fr wfr 5575   ∧ w-bnj17 34876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1094  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-ss 3907  df-fr 5578  df-bnj17 34877

This theorem is referenced by:  bnj1190  35197