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Theorem trclfvg 13604
 Description: The value of the transitive closure of a relation is a superset or (for proper classes) the empty set. (Contributed by RP, 8-May-2020.)
Assertion
Ref Expression
trclfvg (𝑅 ⊆ (t+‘𝑅) ∨ (t+‘𝑅) = ∅)

Proof of Theorem trclfvg
StepHypRef Expression
1 exmid 430 . 2 (𝑅 ∈ V ∨ ¬ 𝑅 ∈ V)
2 trclfvlb 13597 . . 3 (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
3 fvprc 6097 . . 3 𝑅 ∈ V → (t+‘𝑅) = ∅)
42, 3orim12i 537 . 2 ((𝑅 ∈ V ∨ ¬ 𝑅 ∈ V) → (𝑅 ⊆ (t+‘𝑅) ∨ (t+‘𝑅) = ∅))
51, 4ax-mp 5 1 (𝑅 ⊆ (t+‘𝑅) ∨ (t+‘𝑅) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 382   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ‘cfv 5804  t+ctcl 13572 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-trcl 13574 This theorem is referenced by: (None)
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