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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 12509 | . . . 4 ⊢ 0 ∈ ℕ0 | |
2 | 1 | fconst6 6781 | . . 3 ⊢ (𝐼 × {0}):𝐼⟶ℕ0 |
3 | c0ex 11230 | . . . . . 6 ⊢ 0 ∈ V | |
4 | 3 | fconst 6777 | . . . . 5 ⊢ (𝐼 × {0}):𝐼⟶{0} |
5 | incom 4197 | . . . . . 6 ⊢ ({0} ∩ ℕ) = (ℕ ∩ {0}) | |
6 | 0nnn 12270 | . . . . . . 7 ⊢ ¬ 0 ∈ ℕ | |
7 | disjsn 4711 | . . . . . . 7 ⊢ ((ℕ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ) | |
8 | 6, 7 | mpbir 230 | . . . . . 6 ⊢ (ℕ ∩ {0}) = ∅ |
9 | 5, 8 | eqtri 2755 | . . . . 5 ⊢ ({0} ∩ ℕ) = ∅ |
10 | fimacnvdisj 6769 | . . . . 5 ⊢ (((𝐼 × {0}):𝐼⟶{0} ∧ ({0} ∩ ℕ) = ∅) → (◡(𝐼 × {0}) “ ℕ) = ∅) | |
11 | 4, 9, 10 | mp2an 691 | . . . 4 ⊢ (◡(𝐼 × {0}) “ ℕ) = ∅ |
12 | 0fin 9187 | . . . 4 ⊢ ∅ ∈ Fin | |
13 | 11, 12 | eqeltri 2824 | . . 3 ⊢ (◡(𝐼 × {0}) “ ℕ) ∈ Fin |
14 | 2, 13 | pm3.2i 470 | . 2 ⊢ ((𝐼 × {0}):𝐼⟶ℕ0 ∧ (◡(𝐼 × {0}) “ ℕ) ∈ Fin) |
15 | psrbag0.d | . . 3 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
16 | 15 | psrbag 21837 | . 2 ⊢ (𝐼 ∈ 𝑉 → ((𝐼 × {0}) ∈ 𝐷 ↔ ((𝐼 × {0}):𝐼⟶ℕ0 ∧ (◡(𝐼 × {0}) “ ℕ) ∈ Fin))) |
17 | 14, 16 | mpbiri 258 | 1 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {crab 3427 ∩ cin 3943 ∅c0 4318 {csn 4624 × cxp 5670 ◡ccnv 5671 “ cima 5675 ⟶wf 6538 (class class class)co 7414 ↑m cmap 8836 Fincfn 8955 0cc0 11130 ℕcn 12234 ℕ0cn0 12494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 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-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 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 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-nn 12235 df-n0 12495 |
This theorem is referenced by: mplascl 21995 subrgasclcl 21998 evlslem1 22015 tdeglem4 25982 tdeglem4OLD 25983 mdegle0 26000 |
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