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Theorem frirr 5525
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 483 . . 3 ((𝑅 Fr 𝐴𝐵𝐴) → 𝑅 Fr 𝐴)
2 snssi 4733 . . . 4 (𝐵𝐴 → {𝐵} ⊆ 𝐴)
32adantl 482 . . 3 ((𝑅 Fr 𝐴𝐵𝐴) → {𝐵} ⊆ 𝐴)
4 snnzg 4702 . . . 4 (𝐵𝐴 → {𝐵} ≠ ∅)
54adantl 482 . . 3 ((𝑅 Fr 𝐴𝐵𝐴) → {𝐵} ≠ ∅)
6 snex 5322 . . . 4 {𝐵} ∈ V
76frc 5514 . . 3 ((𝑅 Fr 𝐴 ∧ {𝐵} ⊆ 𝐴 ∧ {𝐵} ≠ ∅) → ∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅)
81, 3, 5, 7syl3anc 1363 . 2 ((𝑅 Fr 𝐴𝐵𝐴) → ∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅)
9 rabeq0 4335 . . . . . 6 ({𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝑦)
10 breq2 5061 . . . . . . . 8 (𝑦 = 𝐵 → (𝑥𝑅𝑦𝑥𝑅𝐵))
1110notbid 319 . . . . . . 7 (𝑦 = 𝐵 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝐵))
1211ralbidv 3194 . . . . . 6 (𝑦 = 𝐵 → (∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵))
139, 12syl5bb 284 . . . . 5 (𝑦 = 𝐵 → ({𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵))
1413rexsng 4606 . . . 4 (𝐵𝐴 → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵))
15 breq1 5060 . . . . . 6 (𝑥 = 𝐵 → (𝑥𝑅𝐵𝐵𝑅𝐵))
1615notbid 319 . . . . 5 (𝑥 = 𝐵 → (¬ 𝑥𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵))
1716ralsng 4605 . . . 4 (𝐵𝐴 → (∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵))
1814, 17bitrd 280 . . 3 (𝐵𝐴 → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ¬ 𝐵𝑅𝐵))
1918adantl 482 . 2 ((𝑅 Fr 𝐴𝐵𝐴) → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ¬ 𝐵𝑅𝐵))
208, 19mpbid 233 1 ((𝑅 Fr 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  {crab 3139  wss 3933  c0 4288  {csn 4557   class class class wbr 5057   Fr wfr 5504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-fr 5507
This theorem is referenced by:  efrirr  5529  predfrirr  6170  dfwe2  7485  bnj1417  32210  efrunt  32836  ifr0  40659
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