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Mirrors > Home > MPE Home > Th. List > ex-in | Structured version Visualization version GIF version |
Description: Example for df-in 3922. 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 4594 | . . 3 ⊢ {1, 8} = ({1} ∪ {8}) | |
2 | 1 | ineq2i 4174 | . 2 ⊢ ({1, 3} ∩ {1, 8}) = ({1, 3} ∩ ({1} ∪ {8})) |
3 | indi 4238 | . . 3 ⊢ ({1, 3} ∩ ({1} ∪ {8})) = (({1, 3} ∩ {1}) ∪ ({1, 3} ∩ {8})) | |
4 | snsspr1 4779 | . . . . . 6 ⊢ {1} ⊆ {1, 3} | |
5 | sseqin2 4180 | . . . . . 6 ⊢ ({1} ⊆ {1, 3} ↔ ({1, 3} ∩ {1}) = {1}) | |
6 | 4, 5 | mpbi 229 | . . . . 5 ⊢ ({1, 3} ∩ {1}) = {1} |
7 | 1re 11162 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
8 | 1lt8 12358 | . . . . . . . 8 ⊢ 1 < 8 | |
9 | 7, 8 | gtneii 11274 | . . . . . . 7 ⊢ 8 ≠ 1 |
10 | 3re 12240 | . . . . . . . 8 ⊢ 3 ∈ ℝ | |
11 | 3lt8 12356 | . . . . . . . 8 ⊢ 3 < 8 | |
12 | 10, 11 | gtneii 11274 | . . . . . . 7 ⊢ 8 ≠ 3 |
13 | 9, 12 | nelpri 4620 | . . . . . 6 ⊢ ¬ 8 ∈ {1, 3} |
14 | disjsn 4677 | . . . . . 6 ⊢ (({1, 3} ∩ {8}) = ∅ ↔ ¬ 8 ∈ {1, 3}) | |
15 | 13, 14 | mpbir 230 | . . . . 5 ⊢ ({1, 3} ∩ {8}) = ∅ |
16 | 6, 15 | uneq12i 4126 | . . . 4 ⊢ (({1, 3} ∩ {1}) ∪ ({1, 3} ∩ {8})) = ({1} ∪ ∅) |
17 | un0 4355 | . . . 4 ⊢ ({1} ∪ ∅) = {1} | |
18 | 16, 17 | eqtri 2765 | . . 3 ⊢ (({1, 3} ∩ {1}) ∪ ({1, 3} ∩ {8})) = {1} |
19 | 3, 18 | eqtri 2765 | . 2 ⊢ ({1, 3} ∩ ({1} ∪ {8})) = {1} |
20 | 2, 19 | eqtri 2765 | 1 ⊢ ({1, 3} ∩ {1, 8}) = {1} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 ∪ cun 3913 ∩ cin 3914 ⊆ wss 3915 ∅c0 4287 {csn 4591 {cpr 4593 1c1 11059 3c3 12216 8c8 12221 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-po 5550 df-so 5551 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 |
This theorem is referenced by: (None) |
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