vtxval0 - Intuitionistic Logic Explorer

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Theorem vtxval0 15977

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 4759	. . 3 ⊢ ¬ ∅ ∈ (V × V)
2	1	iffalsei 3618	. 2 ⊢ if(∅ ∈ (V × V), (1^st ‘∅), (Base‘∅)) = (Base‘∅)
3		0ex 4221	. . 3 ⊢ ∅ ∈ V
4		vtxvalg 15940	. . 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 13195	. 2 ⊢ ∅ = (Base‘∅)
7	2, 5, 6	3eqtr4i 2262	1 ⊢ (Vtx‘∅) = ∅

Colors of variables: wff set class

Syntax hints: = wceq 1398 ∈ wcel 2202 Vcvv 2803 ∅c0 3496 ifcif 3607 × cxp 4729 ‘cfv 5333 1^st c1st 6310 Basecbs 13145 Vtxcvtx 15936

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 2204 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-cnex 8166 ax-resscn 8167 ax-1re 8169 ax-addrcl 8172

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-ral 2516 df-rex 2517 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-br 4094 df-opab 4156 df-mpt 4157 df-id 4396 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-fo 5339 df-fv 5341 df-1st 6312 df-inn 9186 df-ndx 13148 df-slot 13149 df-base 13151 df-vtx 15938

This theorem is referenced by: uhgr0 16009 usgr0 16163 0grsubgr 16188 vtxdg0v 16218 0wlk0 16295