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Mirrors > Home > MPE Home > Th. List > ex-in | Structured version Visualization version GIF version |
Description: Example for df-in 3888. 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 4528 | . . 3 ⊢ {1, 8} = ({1} ∪ {8}) | |
2 | 1 | ineq2i 4136 | . 2 ⊢ ({1, 3} ∩ {1, 8}) = ({1, 3} ∩ ({1} ∪ {8})) |
3 | indi 4200 | . . 3 ⊢ ({1, 3} ∩ ({1} ∪ {8})) = (({1, 3} ∩ {1}) ∪ ({1, 3} ∩ {8})) | |
4 | snsspr1 4707 | . . . . . 6 ⊢ {1} ⊆ {1, 3} | |
5 | sseqin2 4142 | . . . . . 6 ⊢ ({1} ⊆ {1, 3} ↔ ({1, 3} ∩ {1}) = {1}) | |
6 | 4, 5 | mpbi 233 | . . . . 5 ⊢ ({1, 3} ∩ {1}) = {1} |
7 | 1re 10630 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
8 | 1lt8 11823 | . . . . . . . 8 ⊢ 1 < 8 | |
9 | 7, 8 | gtneii 10741 | . . . . . . 7 ⊢ 8 ≠ 1 |
10 | 3re 11705 | . . . . . . . 8 ⊢ 3 ∈ ℝ | |
11 | 3lt8 11821 | . . . . . . . 8 ⊢ 3 < 8 | |
12 | 10, 11 | gtneii 10741 | . . . . . . 7 ⊢ 8 ≠ 3 |
13 | 9, 12 | nelpri 4554 | . . . . . 6 ⊢ ¬ 8 ∈ {1, 3} |
14 | disjsn 4607 | . . . . . 6 ⊢ (({1, 3} ∩ {8}) = ∅ ↔ ¬ 8 ∈ {1, 3}) | |
15 | 13, 14 | mpbir 234 | . . . . 5 ⊢ ({1, 3} ∩ {8}) = ∅ |
16 | 6, 15 | uneq12i 4088 | . . . 4 ⊢ (({1, 3} ∩ {1}) ∪ ({1, 3} ∩ {8})) = ({1} ∪ ∅) |
17 | un0 4298 | . . . 4 ⊢ ({1} ∪ ∅) = {1} | |
18 | 16, 17 | eqtri 2821 | . . 3 ⊢ (({1, 3} ∩ {1}) ∪ ({1, 3} ∩ {8})) = {1} |
19 | 3, 18 | eqtri 2821 | . 2 ⊢ ({1, 3} ∩ ({1} ∪ {8})) = {1} |
20 | 2, 19 | eqtri 2821 | 1 ⊢ ({1, 3} ∩ {1, 8}) = {1} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 ∪ cun 3879 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 {csn 4525 {cpr 4527 1c1 10527 3c3 11681 8c8 11686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 |
This theorem is referenced by: (None) |
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