Theorem bnj1154 35168

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 34920	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ∧ 𝐵 ∈ V) → (𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅))
2		elisset 2819	. . . . 5 ⊢ (𝐵 ∈ V → ∃𝑏 𝑏 = 𝐵)
3	2	bnj708 34925	. . . 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 216	. . . . . . 7 ⊢ (𝑅 Fr 𝐴 → ∀𝑏((𝑏 ⊆ 𝐴 ∧ 𝑏 ≠ ∅) → ∃𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ¬ 𝑦𝑅𝑥))
6	5	19.21bi 2197	. . . . . 6 ⊢ (𝑅 Fr 𝐴 → ((𝑏 ⊆ 𝐴 ∧ 𝑏 ≠ ∅) → ∃𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ¬ 𝑦𝑅𝑥))
7	6	3impib 1117	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑏 ⊆ 𝐴 ∧ 𝑏 ≠ ∅) → ∃𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ¬ 𝑦𝑅𝑥)
8		sseq1 3960	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
9		neeq1 2995	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏 ≠ ∅ ↔ 𝐵 ≠ ∅))
10	8, 9	3anbi23d 1442	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝑅 Fr 𝐴 ∧ 𝑏 ⊆ 𝐴 ∧ 𝑏 ≠ ∅) ↔ (𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)))
11		raleq 3294	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
12	11	rexeqbi1dv 3310	. . . . . 6 ⊢ (𝑏 = 𝐵 → (∃𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
13	10, 12	imbi12d 344	. . . . 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 233	. . . 4 ⊢ (𝑏 = 𝐵 → ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
15	3, 14	bnj593 34914	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ∧ 𝐵 ∈ V) → ∃𝑏((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
16	15	bnj937 34940	. 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 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  Vcvv 3441   ⊆ wss 3902  ∅c0 4286   class class class wbr 5099   Fr wfr 5575   ∧ w-bnj17 34855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3062  df-ss 3919  df-fr 5578  df-bnj17 34856

This theorem is referenced by:  bnj1190  35177