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Theorem bnj1154 34992
Description: Property of Fr. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj1154 ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅ ∧ 𝐵 ∈ V) → ∃𝑥𝐵𝑦𝐵 ¬ 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.
StepHypRef Expression
1 bnj658 34744 . 2 ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅ ∧ 𝐵 ∈ V) → (𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅))
2 elisset 2821 . . . . 5 (𝐵 ∈ V → ∃𝑏 𝑏 = 𝐵)
32bnj708 34749 . . . 4 ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅ ∧ 𝐵 ∈ V) → ∃𝑏 𝑏 = 𝐵)
4 df-fr 5641 . . . . . . . 8 (𝑅 Fr 𝐴 ↔ ∀𝑏((𝑏𝐴𝑏 ≠ ∅) → ∃𝑥𝑏𝑦𝑏 ¬ 𝑦𝑅𝑥))
54biimpi 216 . . . . . . 7 (𝑅 Fr 𝐴 → ∀𝑏((𝑏𝐴𝑏 ≠ ∅) → ∃𝑥𝑏𝑦𝑏 ¬ 𝑦𝑅𝑥))
6519.21bi 2187 . . . . . 6 (𝑅 Fr 𝐴 → ((𝑏𝐴𝑏 ≠ ∅) → ∃𝑥𝑏𝑦𝑏 ¬ 𝑦𝑅𝑥))
763impib 1115 . . . . 5 ((𝑅 Fr 𝐴𝑏𝐴𝑏 ≠ ∅) → ∃𝑥𝑏𝑦𝑏 ¬ 𝑦𝑅𝑥)
8 sseq1 4021 . . . . . . 7 (𝑏 = 𝐵 → (𝑏𝐴𝐵𝐴))
9 neeq1 3001 . . . . . . 7 (𝑏 = 𝐵 → (𝑏 ≠ ∅ ↔ 𝐵 ≠ ∅))
108, 93anbi23d 1438 . . . . . 6 (𝑏 = 𝐵 → ((𝑅 Fr 𝐴𝑏𝐴𝑏 ≠ ∅) ↔ (𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅)))
11 raleq 3321 . . . . . . 7 (𝑏 = 𝐵 → (∀𝑦𝑏 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥))
1211rexeqbi1dv 3337 . . . . . 6 (𝑏 = 𝐵 → (∃𝑥𝑏𝑦𝑏 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥))
1310, 12imbi12d 344 . . . . 5 (𝑏 = 𝐵 → (((𝑅 Fr 𝐴𝑏𝐴𝑏 ≠ ∅) → ∃𝑥𝑏𝑦𝑏 ¬ 𝑦𝑅𝑥) ↔ ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)))
147, 13mpbii 233 . . . 4 (𝑏 = 𝐵 → ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥))
153, 14bnj593 34738 . . 3 ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅ ∧ 𝐵 ∈ V) → ∃𝑏((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥))
1615bnj937 34764 . 2 ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅ ∧ 𝐵 ∈ V) → ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥))
171, 16mpd 15 1 ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅ ∧ 𝐵 ∈ V) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086  wal 1535   = wceq 1537  wex 1776  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478  wss 3963  c0 4339   class class class wbr 5148   Fr wfr 5638  w-bnj17 34679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-ss 3980  df-fr 5641  df-bnj17 34680
This theorem is referenced by:  bnj1190  35001
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