vtxval0 - Intuitionistic Logic Explorer

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Theorem vtxval0 16048

Description: Degenerated case 1 for vertices: The set of vertices of the empty set is the empty set. (Contributed by AV, 24-Sep-2020.)

**Assertion**
Ref	Expression
vtxval0	⊢ (Vtx‘∅) = ∅

**Proof of Theorem vtxval0**
Step	Hyp	Ref	Expression
1		0nelxp 4777	. . 3 ⊢ ¬ ∅ ∈ (V × V)
2	1	iffalsei 3631	. 2 ⊢ if(∅ ∈ (V × V), (1^st ‘∅), (Base‘∅)) = (Base‘∅)
3		0ex 4237	. . 3 ⊢ ∅ ∈ V
4		vtxvalg 16011	. . 3 ⊢ (∅ ∈ V → (Vtx‘∅) = if(∅ ∈ (V × V), (1^st ‘∅), (Base‘∅)))
5	3, 4	ax-mp 5	. 2 ⊢ (Vtx‘∅) = if(∅ ∈ (V × V), (1^st ‘∅), (Base‘∅))
6		base0 13262	. 2 ⊢ ∅ = (Base‘∅)
7	2, 5, 6	3eqtr4i 2263	1 ⊢ (Vtx‘∅) = ∅

Colors of variables: wff set class

Syntax hints: = wceq 1398 ∈ wcel 2203 Vcvv 2813 ∅c0 3508 ifcif 3620 × cxp 4747 ‘cfv 5352 1^st c1st 6332 Basecbs 13212 Vtxcvtx 16007

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-cnex 8218 ax-resscn 8219 ax-1re 8221 ax-addrcl 8224

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-ral 2525 df-rex 2526 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-if 3621 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-br 4110 df-opab 4172 df-mpt 4173 df-id 4414 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-fo 5358 df-fv 5360 df-1st 6334 df-inn 9238 df-ndx 13215 df-slot 13216 df-base 13218 df-vtx 16009

This theorem is referenced by: uhgr0 16080 usgr0 16234 0grsubgr 16259 vtxdg0v 16289 0wlk0 16366