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| Description: Example for df-in 3957. 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 4628 | . . 3 ⊢ {1, 8} = ({1} ∪ {8}) | |
| 2 | 1 | ineq2i 4216 | . 2 ⊢ ({1, 3} ∩ {1, 8}) = ({1, 3} ∩ ({1} ∪ {8})) | 
| 3 | indi 4283 | . . 3 ⊢ ({1, 3} ∩ ({1} ∪ {8})) = (({1, 3} ∩ {1}) ∪ ({1, 3} ∩ {8})) | |
| 4 | snsspr1 4813 | . . . . . 6 ⊢ {1} ⊆ {1, 3} | |
| 5 | sseqin2 4222 | . . . . . 6 ⊢ ({1} ⊆ {1, 3} ↔ ({1, 3} ∩ {1}) = {1}) | |
| 6 | 4, 5 | mpbi 230 | . . . . 5 ⊢ ({1, 3} ∩ {1}) = {1} | 
| 7 | 1re 11262 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
| 8 | 1lt8 12465 | . . . . . . . 8 ⊢ 1 < 8 | |
| 9 | 7, 8 | gtneii 11374 | . . . . . . 7 ⊢ 8 ≠ 1 | 
| 10 | 3re 12347 | . . . . . . . 8 ⊢ 3 ∈ ℝ | |
| 11 | 3lt8 12463 | . . . . . . . 8 ⊢ 3 < 8 | |
| 12 | 10, 11 | gtneii 11374 | . . . . . . 7 ⊢ 8 ≠ 3 | 
| 13 | 9, 12 | nelpri 4654 | . . . . . 6 ⊢ ¬ 8 ∈ {1, 3} | 
| 14 | disjsn 4710 | . . . . . 6 ⊢ (({1, 3} ∩ {8}) = ∅ ↔ ¬ 8 ∈ {1, 3}) | |
| 15 | 13, 14 | mpbir 231 | . . . . 5 ⊢ ({1, 3} ∩ {8}) = ∅ | 
| 16 | 6, 15 | uneq12i 4165 | . . . 4 ⊢ (({1, 3} ∩ {1}) ∪ ({1, 3} ∩ {8})) = ({1} ∪ ∅) | 
| 17 | un0 4393 | . . . 4 ⊢ ({1} ∪ ∅) = {1} | |
| 18 | 16, 17 | eqtri 2764 | . . 3 ⊢ (({1, 3} ∩ {1}) ∪ ({1, 3} ∩ {8})) = {1} | 
| 19 | 3, 18 | eqtri 2764 | . 2 ⊢ ({1, 3} ∩ ({1} ∪ {8})) = {1} | 
| 20 | 2, 19 | eqtri 2764 | 1 ⊢ ({1, 3} ∩ {1, 8}) = {1} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 = wceq 1539 ∈ wcel 2107 ∪ cun 3948 ∩ cin 3949 ⊆ wss 3950 ∅c0 4332 {csn 4625 {cpr 4627 1c1 11157 3c3 12323 8c8 12328 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 | 
| This theorem is referenced by: (None) | 
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