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Mirrors > Home > MPE Home > Th. List > ex-in | Structured version Visualization version GIF version |
Description: Example for df-in 3614. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
ex-in | ⊢ ({1, 3} ∩ {1, 8}) = {1} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4213 | . . 3 ⊢ {1, 8} = ({1} ∪ {8}) | |
2 | 1 | ineq2i 3844 | . 2 ⊢ ({1, 3} ∩ {1, 8}) = ({1, 3} ∩ ({1} ∪ {8})) |
3 | indi 3906 | . . 3 ⊢ ({1, 3} ∩ ({1} ∪ {8})) = (({1, 3} ∩ {1}) ∪ ({1, 3} ∩ {8})) | |
4 | snsspr1 4377 | . . . . . 6 ⊢ {1} ⊆ {1, 3} | |
5 | sseqin2 3850 | . . . . . 6 ⊢ ({1} ⊆ {1, 3} ↔ ({1, 3} ∩ {1}) = {1}) | |
6 | 4, 5 | mpbi 220 | . . . . 5 ⊢ ({1, 3} ∩ {1}) = {1} |
7 | 1re 10077 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
8 | 1lt8 11259 | . . . . . . . 8 ⊢ 1 < 8 | |
9 | 7, 8 | gtneii 10187 | . . . . . . 7 ⊢ 8 ≠ 1 |
10 | 3re 11132 | . . . . . . . 8 ⊢ 3 ∈ ℝ | |
11 | 3lt8 11257 | . . . . . . . 8 ⊢ 3 < 8 | |
12 | 10, 11 | gtneii 10187 | . . . . . . 7 ⊢ 8 ≠ 3 |
13 | 9, 12 | nelpri 4234 | . . . . . 6 ⊢ ¬ 8 ∈ {1, 3} |
14 | disjsn 4278 | . . . . . 6 ⊢ (({1, 3} ∩ {8}) = ∅ ↔ ¬ 8 ∈ {1, 3}) | |
15 | 13, 14 | mpbir 221 | . . . . 5 ⊢ ({1, 3} ∩ {8}) = ∅ |
16 | 6, 15 | uneq12i 3798 | . . . 4 ⊢ (({1, 3} ∩ {1}) ∪ ({1, 3} ∩ {8})) = ({1} ∪ ∅) |
17 | un0 4000 | . . . 4 ⊢ ({1} ∪ ∅) = {1} | |
18 | 16, 17 | eqtri 2673 | . . 3 ⊢ (({1, 3} ∩ {1}) ∪ ({1, 3} ∩ {8})) = {1} |
19 | 3, 18 | eqtri 2673 | . 2 ⊢ ({1, 3} ∩ ({1} ∪ {8})) = {1} |
20 | 2, 19 | eqtri 2673 | 1 ⊢ ({1, 3} ∩ {1, 8}) = {1} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1523 ∈ wcel 2030 ∪ cun 3605 ∩ cin 3606 ⊆ wss 3607 ∅c0 3948 {csn 4210 {cpr 4212 1c1 9975 3c3 11109 8c8 11114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 |
This theorem is referenced by: (None) |
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