Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > hashbc0 | Structured version Visualization version GIF version |
Description: The set of subsets of size zero is the singleton of the empty set. (Contributed by Mario Carneiro, 22-Apr-2015.) |
Ref | Expression |
---|---|
ramval.c | ⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖}) |
Ref | Expression |
---|---|
hashbc0 | ⊢ (𝐴 ∈ 𝑉 → (𝐴𝐶0) = {∅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nn0 11307 | . . 3 ⊢ 0 ∈ ℕ0 | |
2 | ramval.c | . . . 4 ⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖}) | |
3 | 2 | hashbcval 15706 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 0 ∈ ℕ0) → (𝐴𝐶0) = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 0}) |
4 | 1, 3 | mpan2 707 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴𝐶0) = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 0}) |
5 | vex 3203 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
6 | hasheq0 13154 | . . . . . . 7 ⊢ (𝑥 ∈ V → ((#‘𝑥) = 0 ↔ 𝑥 = ∅)) | |
7 | 5, 6 | ax-mp 5 | . . . . . 6 ⊢ ((#‘𝑥) = 0 ↔ 𝑥 = ∅) |
8 | 7 | anbi2i 730 | . . . . 5 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ (#‘𝑥) = 0) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 = ∅)) |
9 | id 22 | . . . . . . 7 ⊢ (𝑥 = ∅ → 𝑥 = ∅) | |
10 | 0elpw 4834 | . . . . . . 7 ⊢ ∅ ∈ 𝒫 𝐴 | |
11 | 9, 10 | syl6eqel 2709 | . . . . . 6 ⊢ (𝑥 = ∅ → 𝑥 ∈ 𝒫 𝐴) |
12 | 11 | pm4.71ri 665 | . . . . 5 ⊢ (𝑥 = ∅ ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 = ∅)) |
13 | 8, 12 | bitr4i 267 | . . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ (#‘𝑥) = 0) ↔ 𝑥 = ∅) |
14 | 13 | abbii 2739 | . . 3 ⊢ {𝑥 ∣ (𝑥 ∈ 𝒫 𝐴 ∧ (#‘𝑥) = 0)} = {𝑥 ∣ 𝑥 = ∅} |
15 | df-rab 2921 | . . 3 ⊢ {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 0} = {𝑥 ∣ (𝑥 ∈ 𝒫 𝐴 ∧ (#‘𝑥) = 0)} | |
16 | df-sn 4178 | . . 3 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅} | |
17 | 14, 15, 16 | 3eqtr4i 2654 | . 2 ⊢ {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 0} = {∅} |
18 | 4, 17 | syl6eq 2672 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴𝐶0) = {∅}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {cab 2608 {crab 2916 Vcvv 3200 ∅c0 3915 𝒫 cpw 4158 {csn 4177 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 0cc0 9936 ℕ0cn0 11292 #chash 13117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-hash 13118 |
This theorem is referenced by: 0ram 15724 |
Copyright terms: Public domain | W3C validator |