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Mirrors > Home > MPE Home > Th. List > structvtxvallem | Structured version Visualization version GIF version |
Description: Lemma for structvtxval 25910 and structiedg0val 25911. (Contributed by AV, 23-Sep-2020.) (Revised by AV, 12-Nov-2021.) |
Ref | Expression |
---|---|
structvtxvallem.s | ⊢ 𝑆 ∈ ℕ |
structvtxvallem.b | ⊢ (Base‘ndx) < 𝑆 |
structvtxvallem.g | ⊢ 𝐺 = {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} |
Ref | Expression |
---|---|
structvtxvallem | ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 2 ≤ (#‘dom 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvexd 6203 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Base‘ndx) ∈ V) | |
2 | structvtxvallem.s | . . 3 ⊢ 𝑆 ∈ ℕ | |
3 | 2 | a1i 11 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝑆 ∈ ℕ) |
4 | simpl 473 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝑉 ∈ 𝑋) | |
5 | simpr 477 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝐸 ∈ 𝑌) | |
6 | structvtxvallem.g | . . . 4 ⊢ 𝐺 = {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} | |
7 | prex 4909 | . . . 4 ⊢ {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} ∈ V | |
8 | 6, 7 | eqeltri 2697 | . . 3 ⊢ 𝐺 ∈ V |
9 | 8 | a1i 11 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝐺 ∈ V) |
10 | basendxnn 15924 | . . . . 5 ⊢ (Base‘ndx) ∈ ℕ | |
11 | 10 | nnrei 11029 | . . . 4 ⊢ (Base‘ndx) ∈ ℝ |
12 | structvtxvallem.b | . . . 4 ⊢ (Base‘ndx) < 𝑆 | |
13 | 11, 12 | ltneii 10150 | . . 3 ⊢ (Base‘ndx) ≠ 𝑆 |
14 | 13 | a1i 11 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Base‘ndx) ≠ 𝑆) |
15 | 6 | eqimss2i 3660 | . . 3 ⊢ {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} ⊆ 𝐺 |
16 | 15 | a1i 11 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} ⊆ 𝐺) |
17 | 1, 3, 4, 5, 9, 14, 16 | hashdmpropge2 13265 | 1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 2 ≤ (#‘dom 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ⊆ wss 3574 {cpr 4179 〈cop 4183 class class class wbr 4653 dom cdm 5114 ‘cfv 5888 < clt 10074 ≤ cle 10075 ℕcn 11020 2c2 11070 #chash 13117 ndxcnx 15854 Basecbs 15857 |
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-2 11079 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-hash 13118 df-ndx 15860 df-slot 15861 df-base 15863 |
This theorem is referenced by: structvtxval 25910 structiedg0val 25911 |
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