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Theorem bnj1154 32328
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 32079 . 2 ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅ ∧ 𝐵 ∈ V) → (𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅))
2 elisset 3491 . . . . 5 (𝐵 ∈ V → ∃𝑏 𝑏 = 𝐵)
32bnj708 32084 . . . 4 ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅ ∧ 𝐵 ∈ V) → ∃𝑏 𝑏 = 𝐵)
4 df-fr 5501 . . . . . . . 8 (𝑅 Fr 𝐴 ↔ ∀𝑏((𝑏𝐴𝑏 ≠ ∅) → ∃𝑥𝑏𝑦𝑏 ¬ 𝑦𝑅𝑥))
54biimpi 219 . . . . . . 7 (𝑅 Fr 𝐴 → ∀𝑏((𝑏𝐴𝑏 ≠ ∅) → ∃𝑥𝑏𝑦𝑏 ¬ 𝑦𝑅𝑥))
6519.21bi 2190 . . . . . 6 (𝑅 Fr 𝐴 → ((𝑏𝐴𝑏 ≠ ∅) → ∃𝑥𝑏𝑦𝑏 ¬ 𝑦𝑅𝑥))
763impib 1113 . . . . 5 ((𝑅 Fr 𝐴𝑏𝐴𝑏 ≠ ∅) → ∃𝑥𝑏𝑦𝑏 ¬ 𝑦𝑅𝑥)
8 sseq1 3978 . . . . . . 7 (𝑏 = 𝐵 → (𝑏𝐴𝐵𝐴))
9 neeq1 3076 . . . . . . 7 (𝑏 = 𝐵 → (𝑏 ≠ ∅ ↔ 𝐵 ≠ ∅))
108, 93anbi23d 1436 . . . . . 6 (𝑏 = 𝐵 → ((𝑅 Fr 𝐴𝑏𝐴𝑏 ≠ ∅) ↔ (𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅)))
11 raleq 3396 . . . . . . 7 (𝑏 = 𝐵 → (∀𝑦𝑏 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥))
1211rexeqbi1dv 3395 . . . . . 6 (𝑏 = 𝐵 → (∃𝑥𝑏𝑦𝑏 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥))
1310, 12imbi12d 348 . . . . 5 (𝑏 = 𝐵 → (((𝑅 Fr 𝐴𝑏𝐴𝑏 ≠ ∅) → ∃𝑥𝑏𝑦𝑏 ¬ 𝑦𝑅𝑥) ↔ ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)))
147, 13mpbii 236 . . . 4 (𝑏 = 𝐵 → ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥))
153, 14bnj593 32073 . . 3 ((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅ ∧ 𝐵 ∈ V) → ∃𝑏((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥))
1615bnj937 32100 . 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 399  w3a 1084  wal 1536   = wceq 1538  wex 1781  wcel 2115  wne 3014  wral 3133  wrex 3134  Vcvv 3480  wss 3919  c0 4276   class class class wbr 5052   Fr wfr 5498  w-bnj17 32013
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-ne 3015  df-ral 3138  df-rex 3139  df-v 3482  df-in 3926  df-ss 3936  df-fr 5501  df-bnj17 32014
This theorem is referenced by:  bnj1190  32337
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