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Mirrors > Home > MPE Home > Th. List > 0hashbc | Structured version Visualization version GIF version |
Description: There are no subsets of the empty set with size greater than zero. (Contributed by Mario Carneiro, 22-Apr-2015.) |
Ref | Expression |
---|---|
ramval.c | ⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖}) |
Ref | Expression |
---|---|
0hashbc | ⊢ (𝑁 ∈ ℕ → (∅𝐶𝑁) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0fin 8740 | . . . 4 ⊢ ∅ ∈ Fin | |
2 | nnnn0 11898 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
3 | ramval.c | . . . . 5 ⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖}) | |
4 | 3 | hashbc2 16336 | . . . 4 ⊢ ((∅ ∈ Fin ∧ 𝑁 ∈ ℕ0) → (♯‘(∅𝐶𝑁)) = ((♯‘∅)C𝑁)) |
5 | 1, 2, 4 | sylancr 589 | . . 3 ⊢ (𝑁 ∈ ℕ → (♯‘(∅𝐶𝑁)) = ((♯‘∅)C𝑁)) |
6 | hash0 13722 | . . . . 5 ⊢ (♯‘∅) = 0 | |
7 | 6 | oveq1i 7160 | . . . 4 ⊢ ((♯‘∅)C𝑁) = (0C𝑁) |
8 | bc0k 13665 | . . . 4 ⊢ (𝑁 ∈ ℕ → (0C𝑁) = 0) | |
9 | 7, 8 | syl5eq 2868 | . . 3 ⊢ (𝑁 ∈ ℕ → ((♯‘∅)C𝑁) = 0) |
10 | 5, 9 | eqtrd 2856 | . 2 ⊢ (𝑁 ∈ ℕ → (♯‘(∅𝐶𝑁)) = 0) |
11 | ovex 7183 | . . 3 ⊢ (∅𝐶𝑁) ∈ V | |
12 | hasheq0 13718 | . . 3 ⊢ ((∅𝐶𝑁) ∈ V → ((♯‘(∅𝐶𝑁)) = 0 ↔ (∅𝐶𝑁) = ∅)) | |
13 | 11, 12 | ax-mp 5 | . 2 ⊢ ((♯‘(∅𝐶𝑁)) = 0 ↔ (∅𝐶𝑁) = ∅) |
14 | 10, 13 | sylib 220 | 1 ⊢ (𝑁 ∈ ℕ → (∅𝐶𝑁) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 {crab 3142 Vcvv 3495 ∅c0 4291 𝒫 cpw 4539 ‘cfv 6350 (class class class)co 7150 ∈ cmpo 7152 Fincfn 8503 0cc0 10531 ℕcn 11632 ℕ0cn0 11891 Ccbc 13656 ♯chash 13684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12887 df-seq 13364 df-fac 13628 df-bc 13657 df-hash 13685 |
This theorem is referenced by: ramz2 16354 |
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