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Mirrors > Home > MPE Home > Th. List > upgr0eopALT | Structured version Visualization version GIF version |
Description: Alternate proof of upgr0eop 26826, using the general theorem gropeld 26745 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 26826). (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 3495 | . . . . . 6 ⊢ 𝑔 ∈ V | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ V) |
3 | simpr 485 | . . . . 5 ⊢ (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → (iEdg‘𝑔) = ∅) | |
4 | 2, 3 | upgr0e 26823 | . . . 4 ⊢ (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph) |
5 | 4 | ax-gen 1787 | . . 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 5202 | . . 3 ⊢ ∅ ∈ V | |
9 | 8 | a1i 11 | . 2 ⊢ (𝑉 ∈ 𝑊 → ∅ ∈ V) |
10 | 6, 7, 9 | gropeld 26745 | 1 ⊢ (𝑉 ∈ 𝑊 → 〈𝑉, ∅〉 ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1526 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∅c0 4288 〈cop 4563 ‘cfv 6348 Vtxcvtx 26708 iEdgciedg 26709 UPGraphcupgr 26792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-i2m1 10593 ax-1ne0 10594 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-1st 7678 df-2nd 7679 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-2 11688 df-vtx 26710 df-iedg 26711 df-upgr 26794 df-umgr 26795 |
This theorem is referenced by: (None) |
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