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Mirrors > Home > MPE Home > Th. List > psrbag0 | Structured version Visualization version GIF version |
Description: The empty bag is a bag. (Contributed by Stefan O'Rear, 9-Mar-2015.) |
Ref | Expression |
---|---|
psrbag0.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
Ref | Expression |
---|---|
psrbag0 | ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) ∈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nn0 11636 | . . . 4 ⊢ 0 ∈ ℕ0 | |
2 | 1 | fconst6 6333 | . . 3 ⊢ (𝐼 × {0}):𝐼⟶ℕ0 |
3 | c0ex 10351 | . . . . . 6 ⊢ 0 ∈ V | |
4 | 3 | fconst 6329 | . . . . 5 ⊢ (𝐼 × {0}):𝐼⟶{0} |
5 | incom 4033 | . . . . . 6 ⊢ ({0} ∩ ℕ) = (ℕ ∩ {0}) | |
6 | 0nnn 11388 | . . . . . . 7 ⊢ ¬ 0 ∈ ℕ | |
7 | disjsn 4466 | . . . . . . 7 ⊢ ((ℕ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ) | |
8 | 6, 7 | mpbir 223 | . . . . . 6 ⊢ (ℕ ∩ {0}) = ∅ |
9 | 5, 8 | eqtri 2850 | . . . . 5 ⊢ ({0} ∩ ℕ) = ∅ |
10 | fimacnvdisj 6321 | . . . . 5 ⊢ (((𝐼 × {0}):𝐼⟶{0} ∧ ({0} ∩ ℕ) = ∅) → (◡(𝐼 × {0}) “ ℕ) = ∅) | |
11 | 4, 9, 10 | mp2an 685 | . . . 4 ⊢ (◡(𝐼 × {0}) “ ℕ) = ∅ |
12 | 0fin 8458 | . . . 4 ⊢ ∅ ∈ Fin | |
13 | 11, 12 | eqeltri 2903 | . . 3 ⊢ (◡(𝐼 × {0}) “ ℕ) ∈ Fin |
14 | 2, 13 | pm3.2i 464 | . 2 ⊢ ((𝐼 × {0}):𝐼⟶ℕ0 ∧ (◡(𝐼 × {0}) “ ℕ) ∈ Fin) |
15 | psrbag0.d | . . 3 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
16 | 15 | psrbag 19726 | . 2 ⊢ (𝐼 ∈ 𝑉 → ((𝐼 × {0}) ∈ 𝐷 ↔ ((𝐼 × {0}):𝐼⟶ℕ0 ∧ (◡(𝐼 × {0}) “ ℕ) ∈ Fin))) |
17 | 14, 16 | mpbiri 250 | 1 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 {crab 3122 ∩ cin 3798 ∅c0 4145 {csn 4398 × cxp 5341 ◡ccnv 5342 “ cima 5346 ⟶wf 6120 (class class class)co 6906 ↑𝑚 cmap 8123 Fincfn 8223 0cc0 10253 ℕcn 11351 ℕ0cn0 11619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-er 8010 df-map 8125 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-nn 11352 df-n0 11620 |
This theorem is referenced by: mplascl 19857 subrgasclcl 19860 evlslem1 19876 tdeglem4 24220 mdegle0 24237 |
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