Theorem bnj1154 35251

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 35004	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ∧ 𝐵 ∈ V) → (𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅))
2		elisset 2838	. . . . 5 ⊢ (𝐵 ∈ V → ∃𝑏 𝑏 = 𝐵)
3	2	bnj708 35009	. . . 4 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ∧ 𝐵 ∈ V) → ∃𝑏 𝑏 = 𝐵)
4		df-fr 5593	. . . . . . . 8 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑏((𝑏 ⊆ 𝐴 ∧ 𝑏 ≠ ∅) → ∃𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ¬ 𝑦𝑅𝑥))
5	4	biimpi 218	. . . . . . 7 ⊢ (𝑅 Fr 𝐴 → ∀𝑏((𝑏 ⊆ 𝐴 ∧ 𝑏 ≠ ∅) → ∃𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ¬ 𝑦𝑅𝑥))
6	5	19.21bi 2218	. . . . . 6 ⊢ (𝑅 Fr 𝐴 → ((𝑏 ⊆ 𝐴 ∧ 𝑏 ≠ ∅) → ∃𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ¬ 𝑦𝑅𝑥))
7	6	3impib 1125	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑏 ⊆ 𝐴 ∧ 𝑏 ≠ ∅) → ∃𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ¬ 𝑦𝑅𝑥)
8		sseq1 3956	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
9		neeq1 3013	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏 ≠ ∅ ↔ 𝐵 ≠ ∅))
10	8, 9	3anbi23d 1454	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝑅 Fr 𝐴 ∧ 𝑏 ⊆ 𝐴 ∧ 𝑏 ≠ ∅) ↔ (𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)))
11		raleq 3311	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
12	11	rexeqbi1dv 3325	. . . . . 6 ⊢ (𝑏 = 𝐵 → (∃𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
13	10, 12	imbi12d 346	. . . . 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 235	. . . 4 ⊢ (𝑏 = 𝐵 → ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
15	3, 14	bnj593 34998	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ∧ 𝐵 ∈ V) → ∃𝑏((𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
16	15	bnj937 35024	. 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 398   ∧ w3a 1095  ∀wal 1552   = wceq 1554  ∃wex 1793   ∈ wcel 2136   ≠ wne 2951  ∀wral 3070  ∃wrex 3080  Vcvv 3448   ⊆ wss 3899  ∅c0 4280   class class class wbr 5094   Fr wfr 5590   ∧ w-bnj17 34939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-12 2206  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-3an 1097  df-tru 1557  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ne 2952  df-ral 3071  df-rex 3081  df-ss 3916  df-fr 5593  df-bnj17 34940

This theorem is referenced by:  bnj1190  35260