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Theorem frirr 5665
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 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 482 . . 3 ((𝑅 Fr 𝐴𝐵𝐴) → 𝑅 Fr 𝐴)
2 snssi 4813 . . . 4 (𝐵𝐴 → {𝐵} ⊆ 𝐴)
32adantl 481 . . 3 ((𝑅 Fr 𝐴𝐵𝐴) → {𝐵} ⊆ 𝐴)
4 snnzg 4779 . . . 4 (𝐵𝐴 → {𝐵} ≠ ∅)
54adantl 481 . . 3 ((𝑅 Fr 𝐴𝐵𝐴) → {𝐵} ≠ ∅)
6 snex 5442 . . . 4 {𝐵} ∈ V
76frc 5652 . . 3 ((𝑅 Fr 𝐴 ∧ {𝐵} ⊆ 𝐴 ∧ {𝐵} ≠ ∅) → ∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅)
81, 3, 5, 7syl3anc 1370 . 2 ((𝑅 Fr 𝐴𝐵𝐴) → ∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅)
9 breq1 5151 . . . . . . 7 (𝑥 = 𝑧 → (𝑥𝑅𝑦𝑧𝑅𝑦))
109rabeq0w 4393 . . . . . 6 ({𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑧 ∈ {𝐵} ¬ 𝑧𝑅𝑦)
11 breq2 5152 . . . . . . . 8 (𝑦 = 𝐵 → (𝑧𝑅𝑦𝑧𝑅𝐵))
1211notbid 318 . . . . . . 7 (𝑦 = 𝐵 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑧𝑅𝐵))
1312ralbidv 3176 . . . . . 6 (𝑦 = 𝐵 → (∀𝑧 ∈ {𝐵} ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐵} ¬ 𝑧𝑅𝐵))
1410, 13bitrid 283 . . . . 5 (𝑦 = 𝐵 → ({𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑧 ∈ {𝐵} ¬ 𝑧𝑅𝐵))
1514rexsng 4681 . . . 4 (𝐵𝐴 → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑧 ∈ {𝐵} ¬ 𝑧𝑅𝐵))
16 breq1 5151 . . . . . 6 (𝑧 = 𝐵 → (𝑧𝑅𝐵𝐵𝑅𝐵))
1716notbid 318 . . . . 5 (𝑧 = 𝐵 → (¬ 𝑧𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵))
1817ralsng 4680 . . . 4 (𝐵𝐴 → (∀𝑧 ∈ {𝐵} ¬ 𝑧𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵))
1915, 18bitrd 279 . . 3 (𝐵𝐴 → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ¬ 𝐵𝑅𝐵))
2019adantl 481 . 2 ((𝑅 Fr 𝐴𝐵𝐴) → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ¬ 𝐵𝑅𝐵))
218, 20mpbid 232 1 ((𝑅 Fr 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  wss 3963  c0 4339  {csn 4631   class class class wbr 5148   Fr wfr 5638
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-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  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-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-fr 5641
This theorem is referenced by:  efrirr  5669  predfrirr  6357  dfwe2  7793  bnj1417  35034  efrunt  35693  ifr0  44446
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