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Mirrors > Home > MPE Home > Th. List > trclfvg | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
trclfvg | ⊢ (𝑅 ⊆ (t+‘𝑅) ∨ (t+‘𝑅) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmid 891 | . 2 ⊢ (𝑅 ∈ V ∨ ¬ 𝑅 ∈ V) | |
2 | trclfvlb 14368 | . . 3 ⊢ (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅)) | |
3 | fvprc 6663 | . . 3 ⊢ (¬ 𝑅 ∈ V → (t+‘𝑅) = ∅) | |
4 | 2, 3 | orim12i 905 | . 2 ⊢ ((𝑅 ∈ V ∨ ¬ 𝑅 ∈ V) → (𝑅 ⊆ (t+‘𝑅) ∨ (t+‘𝑅) = ∅)) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ (𝑅 ⊆ (t+‘𝑅) ∨ (t+‘𝑅) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 843 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 ∅c0 4291 ‘cfv 6355 t+ctcl 14345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-int 4877 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-iota 6314 df-fun 6357 df-fv 6363 df-trcl 14347 |
This theorem is referenced by: (None) |
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