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Mirrors > Home > MPE Home > Th. List > upgr0eopALT | Structured version Visualization version GIF version |
Description: Alternate proof of upgr0eop 28969, using the general theorem gropeld 28888 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 28969). (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 3467 | . . . . . 6 ⊢ 𝑔 ∈ V | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ V) |
3 | simpr 483 | . . . . 5 ⊢ (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → (iEdg‘𝑔) = ∅) | |
4 | 2, 3 | upgr0e 28966 | . . . 4 ⊢ (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph) |
5 | 4 | ax-gen 1789 | . . 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 5302 | . . 3 ⊢ ∅ ∈ V | |
9 | 8 | a1i 11 | . 2 ⊢ (𝑉 ∈ 𝑊 → ∅ ∈ V) |
10 | 6, 7, 9 | gropeld 28888 | 1 ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ∅⟩ ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∀wal 1531 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ∅c0 4318 ⟨cop 4630 ‘cfv 6542 Vtxcvtx 28851 iEdgciedg 28852 UPGraphcupgr 28935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-i2m1 11204 ax-1ne0 11205 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 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 7418 df-1st 7989 df-2nd 7990 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-2 12303 df-vtx 28853 df-iedg 28854 df-upgr 28937 df-umgr 28938 |
This theorem is referenced by: (None) |
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