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Mirrors > Home > MPE Home > Th. List > cusgrexilem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for cusgrexi 29295. (Contributed by AV, 12-Jan-2018.) (Revised by AV, 10-Nov-2021.) |
Ref | Expression |
---|---|
usgrexi.p | ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} |
Ref | Expression |
---|---|
cusgrexilem2 | ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 483 | . . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉) | |
2 | eldifi 4120 | . . . 4 ⊢ (𝑛 ∈ (𝑉 ∖ {𝑣}) → 𝑛 ∈ 𝑉) | |
3 | prelpwi 5444 | . . . 4 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → {𝑣, 𝑛} ∈ 𝒫 𝑉) | |
4 | 1, 2, 3 | syl2an 594 | . . 3 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → {𝑣, 𝑛} ∈ 𝒫 𝑉) |
5 | eldifsni 4790 | . . . . . 6 ⊢ (𝑛 ∈ (𝑉 ∖ {𝑣}) → 𝑛 ≠ 𝑣) | |
6 | 5 | necomd 2986 | . . . . 5 ⊢ (𝑛 ∈ (𝑉 ∖ {𝑣}) → 𝑣 ≠ 𝑛) |
7 | 6 | adantl 480 | . . . 4 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑣 ≠ 𝑛) |
8 | hashprg 14381 | . . . . 5 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → (𝑣 ≠ 𝑛 ↔ (♯‘{𝑣, 𝑛}) = 2)) | |
9 | 1, 2, 8 | syl2an 594 | . . . 4 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (𝑣 ≠ 𝑛 ↔ (♯‘{𝑣, 𝑛}) = 2)) |
10 | 7, 9 | mpbid 231 | . . 3 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (♯‘{𝑣, 𝑛}) = 2) |
11 | fveqeq2 6899 | . . . 4 ⊢ (𝑥 = {𝑣, 𝑛} → ((♯‘𝑥) = 2 ↔ (♯‘{𝑣, 𝑛}) = 2)) | |
12 | rnresi 6074 | . . . . 5 ⊢ ran ( I ↾ 𝑃) = 𝑃 | |
13 | usgrexi.p | . . . . 5 ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} | |
14 | 12, 13 | eqtri 2753 | . . . 4 ⊢ ran ( I ↾ 𝑃) = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} |
15 | 11, 14 | elrab2 3679 | . . 3 ⊢ ({𝑣, 𝑛} ∈ ran ( I ↾ 𝑃) ↔ ({𝑣, 𝑛} ∈ 𝒫 𝑉 ∧ (♯‘{𝑣, 𝑛}) = 2)) |
16 | 4, 10, 15 | sylanbrc 581 | . 2 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → {𝑣, 𝑛} ∈ ran ( I ↾ 𝑃)) |
17 | sseq2 4000 | . . 3 ⊢ (𝑒 = {𝑣, 𝑛} → ({𝑣, 𝑛} ⊆ 𝑒 ↔ {𝑣, 𝑛} ⊆ {𝑣, 𝑛})) | |
18 | 17 | adantl 480 | . 2 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) ∧ 𝑒 = {𝑣, 𝑛}) → ({𝑣, 𝑛} ⊆ 𝑒 ↔ {𝑣, 𝑛} ⊆ {𝑣, 𝑛})) |
19 | ssidd 3997 | . 2 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → {𝑣, 𝑛} ⊆ {𝑣, 𝑛}) | |
20 | 16, 18, 19 | rspcedvd 3605 | 1 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∃wrex 3060 {crab 3419 ∖ cdif 3938 ⊆ wss 3941 𝒫 cpw 4599 {csn 4625 {cpr 4627 I cid 5570 ran crn 5674 ↾ cres 5675 ‘cfv 6543 2c2 12292 ♯chash 14316 |
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 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-dju 9919 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-n0 12498 df-z 12584 df-uz 12848 df-fz 13512 df-hash 14317 |
This theorem is referenced by: cusgrexi 29295 structtocusgr 29298 |
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