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Theorem frirr 5520
Description: A well-founded relation is irreflexive. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 2-Jan-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
Assertion
Ref Expression
frirr ((𝑅 Fr 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)

Proof of Theorem frirr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 486 . . 3 ((𝑅 Fr 𝐴𝐵𝐴) → 𝑅 Fr 𝐴)
2 snssi 4725 . . . 4 (𝐵𝐴 → {𝐵} ⊆ 𝐴)
32adantl 485 . . 3 ((𝑅 Fr 𝐴𝐵𝐴) → {𝐵} ⊆ 𝐴)
4 snnzg 4695 . . . 4 (𝐵𝐴 → {𝐵} ≠ ∅)
54adantl 485 . . 3 ((𝑅 Fr 𝐴𝐵𝐴) → {𝐵} ≠ ∅)
6 snex 5320 . . . 4 {𝐵} ∈ V
76frc 5509 . . 3 ((𝑅 Fr 𝐴 ∧ {𝐵} ⊆ 𝐴 ∧ {𝐵} ≠ ∅) → ∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅)
81, 3, 5, 7syl3anc 1368 . 2 ((𝑅 Fr 𝐴𝐵𝐴) → ∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅)
9 rabeq0 4321 . . . . . 6 ({𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝑦)
10 breq2 5057 . . . . . . . 8 (𝑦 = 𝐵 → (𝑥𝑅𝑦𝑥𝑅𝐵))
1110notbid 321 . . . . . . 7 (𝑦 = 𝐵 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝐵))
1211ralbidv 3192 . . . . . 6 (𝑦 = 𝐵 → (∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵))
139, 12syl5bb 286 . . . . 5 (𝑦 = 𝐵 → ({𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵))
1413rexsng 4599 . . . 4 (𝐵𝐴 → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵))
15 breq1 5056 . . . . . 6 (𝑥 = 𝐵 → (𝑥𝑅𝐵𝐵𝑅𝐵))
1615notbid 321 . . . . 5 (𝑥 = 𝐵 → (¬ 𝑥𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵))
1716ralsng 4598 . . . 4 (𝐵𝐴 → (∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵))
1814, 17bitrd 282 . . 3 (𝐵𝐴 → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ¬ 𝐵𝑅𝐵))
1918adantl 485 . 2 ((𝑅 Fr 𝐴𝐵𝐴) → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ¬ 𝐵𝑅𝐵))
208, 19mpbid 235 1 ((𝑅 Fr 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wne 3014  wral 3133  wrex 3134  {crab 3137  wss 3919  c0 4276  {csn 4550   class class class wbr 5053   Fr wfr 5499
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-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-sn 4551  df-pr 4553  df-op 4557  df-br 5054  df-fr 5502
This theorem is referenced by:  efrirr  5524  predfrirr  6166  dfwe2  7492  bnj1417  32398  efrunt  33024  ifr0  41102
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