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Mirrors > Home > MPE Home > Th. List > 1conngra | Structured version Visualization version GIF version |
Description: A class/graph with (at most) one vertex is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) |
Ref | Expression |
---|---|
1conngra | ⊢ (𝐸 ∈ 𝑉 → {𝐴} ConnGrph 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 476 | . . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝐸 ∈ 𝑉) → 𝐸 ∈ 𝑉) | |
2 | snex 4835 | . . . . . . . 8 ⊢ {𝐴} ∈ V | |
3 | 1, 2 | jctil 558 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐸 ∈ 𝑉) → ({𝐴} ∈ V ∧ 𝐸 ∈ 𝑉)) |
4 | snidg 4153 | . . . . . . . 8 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴}) | |
5 | 4 | adantr 480 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐸 ∈ 𝑉) → 𝐴 ∈ {𝐴}) |
6 | 0pthonv 26111 | . . . . . . 7 ⊢ (({𝐴} ∈ V ∧ 𝐸 ∈ 𝑉) → (𝐴 ∈ {𝐴} → ∃𝑓∃𝑝 𝑓(𝐴({𝐴} PathOn 𝐸)𝐴)𝑝)) | |
7 | 3, 5, 6 | sylc 63 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐸 ∈ 𝑉) → ∃𝑓∃𝑝 𝑓(𝐴({𝐴} PathOn 𝐸)𝐴)𝑝) |
8 | oveq2 6557 | . . . . . . . . . 10 ⊢ (𝑛 = 𝐴 → (𝐴({𝐴} PathOn 𝐸)𝑛) = (𝐴({𝐴} PathOn 𝐸)𝐴)) | |
9 | 8 | breqd 4594 | . . . . . . . . 9 ⊢ (𝑛 = 𝐴 → (𝑓(𝐴({𝐴} PathOn 𝐸)𝑛)𝑝 ↔ 𝑓(𝐴({𝐴} PathOn 𝐸)𝐴)𝑝)) |
10 | 9 | 2exbidv 1839 | . . . . . . . 8 ⊢ (𝑛 = 𝐴 → (∃𝑓∃𝑝 𝑓(𝐴({𝐴} PathOn 𝐸)𝑛)𝑝 ↔ ∃𝑓∃𝑝 𝑓(𝐴({𝐴} PathOn 𝐸)𝐴)𝑝)) |
11 | 10 | ralsng 4165 | . . . . . . 7 ⊢ (𝐴 ∈ V → (∀𝑛 ∈ {𝐴}∃𝑓∃𝑝 𝑓(𝐴({𝐴} PathOn 𝐸)𝑛)𝑝 ↔ ∃𝑓∃𝑝 𝑓(𝐴({𝐴} PathOn 𝐸)𝐴)𝑝)) |
12 | 11 | adantr 480 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐸 ∈ 𝑉) → (∀𝑛 ∈ {𝐴}∃𝑓∃𝑝 𝑓(𝐴({𝐴} PathOn 𝐸)𝑛)𝑝 ↔ ∃𝑓∃𝑝 𝑓(𝐴({𝐴} PathOn 𝐸)𝐴)𝑝)) |
13 | 7, 12 | mpbird 246 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐸 ∈ 𝑉) → ∀𝑛 ∈ {𝐴}∃𝑓∃𝑝 𝑓(𝐴({𝐴} PathOn 𝐸)𝑛)𝑝) |
14 | oveq1 6556 | . . . . . . . . . 10 ⊢ (𝑘 = 𝐴 → (𝑘({𝐴} PathOn 𝐸)𝑛) = (𝐴({𝐴} PathOn 𝐸)𝑛)) | |
15 | 14 | breqd 4594 | . . . . . . . . 9 ⊢ (𝑘 = 𝐴 → (𝑓(𝑘({𝐴} PathOn 𝐸)𝑛)𝑝 ↔ 𝑓(𝐴({𝐴} PathOn 𝐸)𝑛)𝑝)) |
16 | 15 | 2exbidv 1839 | . . . . . . . 8 ⊢ (𝑘 = 𝐴 → (∃𝑓∃𝑝 𝑓(𝑘({𝐴} PathOn 𝐸)𝑛)𝑝 ↔ ∃𝑓∃𝑝 𝑓(𝐴({𝐴} PathOn 𝐸)𝑛)𝑝)) |
17 | 16 | ralbidv 2969 | . . . . . . 7 ⊢ (𝑘 = 𝐴 → (∀𝑛 ∈ {𝐴}∃𝑓∃𝑝 𝑓(𝑘({𝐴} PathOn 𝐸)𝑛)𝑝 ↔ ∀𝑛 ∈ {𝐴}∃𝑓∃𝑝 𝑓(𝐴({𝐴} PathOn 𝐸)𝑛)𝑝)) |
18 | 17 | ralsng 4165 | . . . . . 6 ⊢ (𝐴 ∈ V → (∀𝑘 ∈ {𝐴}∀𝑛 ∈ {𝐴}∃𝑓∃𝑝 𝑓(𝑘({𝐴} PathOn 𝐸)𝑛)𝑝 ↔ ∀𝑛 ∈ {𝐴}∃𝑓∃𝑝 𝑓(𝐴({𝐴} PathOn 𝐸)𝑛)𝑝)) |
19 | 18 | adantr 480 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐸 ∈ 𝑉) → (∀𝑘 ∈ {𝐴}∀𝑛 ∈ {𝐴}∃𝑓∃𝑝 𝑓(𝑘({𝐴} PathOn 𝐸)𝑛)𝑝 ↔ ∀𝑛 ∈ {𝐴}∃𝑓∃𝑝 𝑓(𝐴({𝐴} PathOn 𝐸)𝑛)𝑝)) |
20 | 13, 19 | mpbird 246 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐸 ∈ 𝑉) → ∀𝑘 ∈ {𝐴}∀𝑛 ∈ {𝐴}∃𝑓∃𝑝 𝑓(𝑘({𝐴} PathOn 𝐸)𝑛)𝑝) |
21 | isconngra 26200 | . . . . 5 ⊢ (({𝐴} ∈ V ∧ 𝐸 ∈ 𝑉) → ({𝐴} ConnGrph 𝐸 ↔ ∀𝑘 ∈ {𝐴}∀𝑛 ∈ {𝐴}∃𝑓∃𝑝 𝑓(𝑘({𝐴} PathOn 𝐸)𝑛)𝑝)) | |
22 | 3, 21 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐸 ∈ 𝑉) → ({𝐴} ConnGrph 𝐸 ↔ ∀𝑘 ∈ {𝐴}∀𝑛 ∈ {𝐴}∃𝑓∃𝑝 𝑓(𝑘({𝐴} PathOn 𝐸)𝑛)𝑝)) |
23 | 20, 22 | mpbird 246 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐸 ∈ 𝑉) → {𝐴} ConnGrph 𝐸) |
24 | 23 | ex 449 | . 2 ⊢ (𝐴 ∈ V → (𝐸 ∈ 𝑉 → {𝐴} ConnGrph 𝐸)) |
25 | snprc 4197 | . . 3 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
26 | 0conngra 26202 | . . . 4 ⊢ (𝐸 ∈ 𝑉 → ∅ ConnGrph 𝐸) | |
27 | breq1 4586 | . . . 4 ⊢ ({𝐴} = ∅ → ({𝐴} ConnGrph 𝐸 ↔ ∅ ConnGrph 𝐸)) | |
28 | 26, 27 | syl5ibr 235 | . . 3 ⊢ ({𝐴} = ∅ → (𝐸 ∈ 𝑉 → {𝐴} ConnGrph 𝐸)) |
29 | 25, 28 | sylbi 206 | . 2 ⊢ (¬ 𝐴 ∈ V → (𝐸 ∈ 𝑉 → {𝐴} ConnGrph 𝐸)) |
30 | 24, 29 | pm2.61i 175 | 1 ⊢ (𝐸 ∈ 𝑉 → {𝐴} ConnGrph 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∅c0 3874 {csn 4125 class class class wbr 4583 (class class class)co 6549 PathOn cpthon 26032 ConnGrph cconngra 26197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-wlk 26036 df-trail 26037 df-pth 26038 df-wlkon 26042 df-pthon 26044 df-conngra 26198 |
This theorem is referenced by: (None) |
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