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Mirrors > Home > MPE Home > Th. List > rusgr1vtxlem | Structured version Visualization version GIF version |
Description: Lemma for rusgr1vtx 26484. (Contributed by AV, 27-Dec-2020.) |
Ref | Expression |
---|---|
rusgr1vtxlem | ⊢ (((∀𝑣 ∈ 𝑉 (#‘𝐴) = 𝐾 ∧ ∀𝑣 ∈ 𝑉 𝐴 = ∅) ∧ (𝑉 ∈ 𝑊 ∧ (#‘𝑉) = 1)) → 𝐾 = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.26 3064 | . . 3 ⊢ (∀𝑣 ∈ 𝑉 ((#‘𝐴) = 𝐾 ∧ 𝐴 = ∅) ↔ (∀𝑣 ∈ 𝑉 (#‘𝐴) = 𝐾 ∧ ∀𝑣 ∈ 𝑉 𝐴 = ∅)) | |
2 | fveq2 6191 | . . . . . . 7 ⊢ (𝐴 = ∅ → (#‘𝐴) = (#‘∅)) | |
3 | 2 | eqeq1d 2624 | . . . . . 6 ⊢ (𝐴 = ∅ → ((#‘𝐴) = 𝐾 ↔ (#‘∅) = 𝐾)) |
4 | 3 | biimpac 503 | . . . . 5 ⊢ (((#‘𝐴) = 𝐾 ∧ 𝐴 = ∅) → (#‘∅) = 𝐾) |
5 | 4 | ralimi 2952 | . . . 4 ⊢ (∀𝑣 ∈ 𝑉 ((#‘𝐴) = 𝐾 ∧ 𝐴 = ∅) → ∀𝑣 ∈ 𝑉 (#‘∅) = 𝐾) |
6 | hash1n0 13209 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ (#‘𝑉) = 1) → 𝑉 ≠ ∅) | |
7 | rspn0 3934 | . . . . . 6 ⊢ (𝑉 ≠ ∅ → (∀𝑣 ∈ 𝑉 (#‘∅) = 𝐾 → (#‘∅) = 𝐾)) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ (#‘𝑉) = 1) → (∀𝑣 ∈ 𝑉 (#‘∅) = 𝐾 → (#‘∅) = 𝐾)) |
9 | hash0 13158 | . . . . . 6 ⊢ (#‘∅) = 0 | |
10 | eqeq1 2626 | . . . . . 6 ⊢ ((#‘∅) = 𝐾 → ((#‘∅) = 0 ↔ 𝐾 = 0)) | |
11 | 9, 10 | mpbii 223 | . . . . 5 ⊢ ((#‘∅) = 𝐾 → 𝐾 = 0) |
12 | 8, 11 | syl6com 37 | . . . 4 ⊢ (∀𝑣 ∈ 𝑉 (#‘∅) = 𝐾 → ((𝑉 ∈ 𝑊 ∧ (#‘𝑉) = 1) → 𝐾 = 0)) |
13 | 5, 12 | syl 17 | . . 3 ⊢ (∀𝑣 ∈ 𝑉 ((#‘𝐴) = 𝐾 ∧ 𝐴 = ∅) → ((𝑉 ∈ 𝑊 ∧ (#‘𝑉) = 1) → 𝐾 = 0)) |
14 | 1, 13 | sylbir 225 | . 2 ⊢ ((∀𝑣 ∈ 𝑉 (#‘𝐴) = 𝐾 ∧ ∀𝑣 ∈ 𝑉 𝐴 = ∅) → ((𝑉 ∈ 𝑊 ∧ (#‘𝑉) = 1) → 𝐾 = 0)) |
15 | 14 | imp 445 | 1 ⊢ (((∀𝑣 ∈ 𝑉 (#‘𝐴) = 𝐾 ∧ ∀𝑣 ∈ 𝑉 𝐴 = ∅) ∧ (𝑉 ∈ 𝑊 ∧ (#‘𝑉) = 1)) → 𝐾 = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∅c0 3915 ‘cfv 5888 0cc0 9936 1c1 9937 #chash 13117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-hash 13118 |
This theorem is referenced by: rusgr1vtx 26484 |
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