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Mirrors > Home > MPE Home > Th. List > upgr0eopALT | Structured version Visualization version GIF version |
Description: Alternate proof of upgr0eop 28901, using the general theorem gropeld 28820 to transform a theorem for an arbitrary representation of a graph into a theorem for a graph represented as ordered pair. This general approach causes some overhead, which makes the proof longer than necessary (see proof of upgr0eop 28901). (Contributed by AV, 11-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
upgr0eopALT | ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ∅⟩ ∈ UPGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3473 | . . . . . 6 ⊢ 𝑔 ∈ V | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ V) |
3 | simpr 484 | . . . . 5 ⊢ (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → (iEdg‘𝑔) = ∅) | |
4 | 2, 3 | upgr0e 28898 | . . . 4 ⊢ (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph) |
5 | 4 | ax-gen 1790 | . . 3 ⊢ ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph) |
6 | 5 | a1i 11 | . 2 ⊢ (𝑉 ∈ 𝑊 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph)) |
7 | id 22 | . 2 ⊢ (𝑉 ∈ 𝑊 → 𝑉 ∈ 𝑊) | |
8 | 0ex 5301 | . . 3 ⊢ ∅ ∈ V | |
9 | 8 | a1i 11 | . 2 ⊢ (𝑉 ∈ 𝑊 → ∅ ∈ V) |
10 | 6, 7, 9 | gropeld 28820 | 1 ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ∅⟩ ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1532 = wceq 1534 ∈ wcel 2099 Vcvv 3469 ∅c0 4318 ⟨cop 4630 ‘cfv 6542 Vtxcvtx 28783 iEdgciedg 28784 UPGraphcupgr 28867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-i2m1 11192 ax-1ne0 11193 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-1st 7985 df-2nd 7986 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-2 12291 df-vtx 28785 df-iedg 28786 df-upgr 28869 df-umgr 28870 |
This theorem is referenced by: (None) |
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