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Mirrors > Home > MPE Home > Th. List > upgrbi | Structured version Visualization version GIF version |
Description: Show that an unordered pair is a valid edge in a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 28-Feb-2021.) |
Ref | Expression |
---|---|
upgrbi.x | ⊢ 𝑋 ∈ 𝑉 |
upgrbi.y | ⊢ 𝑌 ∈ 𝑉 |
Ref | Expression |
---|---|
upgrbi | ⊢ {𝑋, 𝑌} ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | upgrbi.x | . . . . 5 ⊢ 𝑋 ∈ 𝑉 | |
2 | upgrbi.y | . . . . 5 ⊢ 𝑌 ∈ 𝑉 | |
3 | prssi 4628 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ⊆ 𝑉) | |
4 | 1, 2, 3 | mp2an 679 | . . . 4 ⊢ {𝑋, 𝑌} ⊆ 𝑉 |
5 | prex 5189 | . . . . 5 ⊢ {𝑋, 𝑌} ∈ V | |
6 | 5 | elpw 4428 | . . . 4 ⊢ ({𝑋, 𝑌} ∈ 𝒫 𝑉 ↔ {𝑋, 𝑌} ⊆ 𝑉) |
7 | 4, 6 | mpbir 223 | . . 3 ⊢ {𝑋, 𝑌} ∈ 𝒫 𝑉 |
8 | 1 | elexi 3435 | . . . 4 ⊢ 𝑋 ∈ V |
9 | 8 | prnz 4586 | . . 3 ⊢ {𝑋, 𝑌} ≠ ∅ |
10 | eldifsn 4593 | . . 3 ⊢ ({𝑋, 𝑌} ∈ (𝒫 𝑉 ∖ {∅}) ↔ ({𝑋, 𝑌} ∈ 𝒫 𝑉 ∧ {𝑋, 𝑌} ≠ ∅)) | |
11 | 7, 9, 10 | mpbir2an 698 | . 2 ⊢ {𝑋, 𝑌} ∈ (𝒫 𝑉 ∖ {∅}) |
12 | hashprlei 13637 | . . 3 ⊢ ({𝑋, 𝑌} ∈ Fin ∧ (♯‘{𝑋, 𝑌}) ≤ 2) | |
13 | 12 | simpri 478 | . 2 ⊢ (♯‘{𝑋, 𝑌}) ≤ 2 |
14 | fveq2 6499 | . . . 4 ⊢ (𝑥 = {𝑋, 𝑌} → (♯‘𝑥) = (♯‘{𝑋, 𝑌})) | |
15 | 14 | breq1d 4939 | . . 3 ⊢ (𝑥 = {𝑋, 𝑌} → ((♯‘𝑥) ≤ 2 ↔ (♯‘{𝑋, 𝑌}) ≤ 2)) |
16 | 15 | elrab 3596 | . 2 ⊢ ({𝑋, 𝑌} ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ ({𝑋, 𝑌} ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘{𝑋, 𝑌}) ≤ 2)) |
17 | 11, 13, 16 | mpbir2an 698 | 1 ⊢ {𝑋, 𝑌} ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ∈ wcel 2050 ≠ wne 2968 {crab 3093 ∖ cdif 3827 ⊆ wss 3830 ∅c0 4179 𝒫 cpw 4422 {csn 4441 {cpr 4443 class class class wbr 4929 ‘cfv 6188 Fincfn 8306 ≤ cle 10475 2c2 11495 ♯chash 13505 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-dju 9124 df-card 9162 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-n0 11708 df-xnn0 11780 df-z 11794 df-uz 12059 df-fz 12709 df-hash 13506 |
This theorem is referenced by: (None) |
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