vtxval0 - Intuitionistic Logic Explorer

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Theorem vtxval0 16208

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 4797	. . 3 ⊢ ¬ ∅ ∈ (V × V)
2	1	iffalsei 3646	. 2 ⊢ if(∅ ∈ (V × V), (1^st ‘∅), (Base‘∅)) = (Base‘∅)
3		0ex 4255	. . 3 ⊢ ∅ ∈ V
4		vtxvalg 16171	. . 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 13380	. 2 ⊢ ∅ = (Base‘∅)
7	2, 5, 6	3eqtr4i 2269	1 ⊢ (Vtx‘∅) = ∅

Colors of variables: wff set class

Syntax hints: = wceq 1402 ∈ wcel 2209 Vcvv 2821 ∅c0 3520 ifcif 3635 × cxp 4767 ‘cfv 5372 1^st c1st 6362 Basecbs 13330 Vtxcvtx 16167

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-cnex 8260 ax-resscn 8261 ax-1re 8263 ax-addrcl 8266

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-br 4126 df-opab 4188 df-mpt 4189 df-id 4433 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-fo 5378 df-fv 5380 df-1st 6364 df-inn 9284 df-ndx 13333 df-slot 13334 df-base 13336 df-vtx 16169

This theorem is referenced by: uhgr0 16240 usgr0 16394 0grsubgr 16419 vtxdg0v 16449 0wlk0 16526