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Mirrors > Home > MPE Home > Th. List > vtxval0 | Structured version Visualization version GIF version |
Description: Degenerated case 1 for vertices: The set of vertices of the empty set is the empty set. (Contributed by AV, 24-Sep-2020.) |
Ref | Expression |
---|---|
vtxval0 | ⊢ (Vtx‘∅) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nelxp 5067 | . . 3 ⊢ ¬ ∅ ∈ (V × V) | |
2 | 1 | iffalsei 4046 | . 2 ⊢ if(∅ ∈ (V × V), (1st ‘∅), (Base‘∅)) = (Base‘∅) |
3 | 0ex 4718 | . . 3 ⊢ ∅ ∈ V | |
4 | vtxval 25677 | . . 3 ⊢ (∅ ∈ V → (Vtx‘∅) = if(∅ ∈ (V × V), (1st ‘∅), (Base‘∅))) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ (Vtx‘∅) = if(∅ ∈ (V × V), (1st ‘∅), (Base‘∅)) |
6 | base0 15740 | . 2 ⊢ ∅ = (Base‘∅) | |
7 | 2, 5, 6 | 3eqtr4i 2642 | 1 ⊢ (Vtx‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ifcif 4036 × cxp 5036 ‘cfv 5804 1st c1st 7057 Basecbs 15695 Vtxcvtx 25673 |
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 |
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-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 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-iota 5768 df-fun 5806 df-fv 5812 df-slot 15699 df-base 15700 df-vtx 25675 |
This theorem is referenced by: uhgr0 25739 usgr0 40469 0grsubgr 40502 cplgr0 40647 vtxdg0v 40688 0grrusgr 40779 01wlk0 40861 0conngr 41359 frgr0 41436 |
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