Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ex-dif | Structured version Visualization version GIF version |
Description: Example for df-dif 3938. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
ex-dif | ⊢ ({1, 3} ∖ {1, 8}) = {3} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4563 | . . 3 ⊢ {1, 3} = ({1} ∪ {3}) | |
2 | 1 | difeq1i 4094 | . 2 ⊢ ({1, 3} ∖ {1, 8}) = (({1} ∪ {3}) ∖ {1, 8}) |
3 | difundir 4256 | . 2 ⊢ (({1} ∪ {3}) ∖ {1, 8}) = (({1} ∖ {1, 8}) ∪ ({3} ∖ {1, 8})) | |
4 | snsspr1 4740 | . . . . 5 ⊢ {1} ⊆ {1, 8} | |
5 | ssdif0 4322 | . . . . 5 ⊢ ({1} ⊆ {1, 8} ↔ ({1} ∖ {1, 8}) = ∅) | |
6 | 4, 5 | mpbi 232 | . . . 4 ⊢ ({1} ∖ {1, 8}) = ∅ |
7 | incom 4177 | . . . . . . 7 ⊢ ({3} ∩ {1, 8}) = ({1, 8} ∩ {3}) | |
8 | 1re 10635 | . . . . . . . . . 10 ⊢ 1 ∈ ℝ | |
9 | 1lt3 11804 | . . . . . . . . . 10 ⊢ 1 < 3 | |
10 | 8, 9 | gtneii 10746 | . . . . . . . . 9 ⊢ 3 ≠ 1 |
11 | 3re 11711 | . . . . . . . . . 10 ⊢ 3 ∈ ℝ | |
12 | 3lt8 11827 | . . . . . . . . . 10 ⊢ 3 < 8 | |
13 | 11, 12 | ltneii 10747 | . . . . . . . . 9 ⊢ 3 ≠ 8 |
14 | 10, 13 | nelpri 4587 | . . . . . . . 8 ⊢ ¬ 3 ∈ {1, 8} |
15 | disjsn 4640 | . . . . . . . 8 ⊢ (({1, 8} ∩ {3}) = ∅ ↔ ¬ 3 ∈ {1, 8}) | |
16 | 14, 15 | mpbir 233 | . . . . . . 7 ⊢ ({1, 8} ∩ {3}) = ∅ |
17 | 7, 16 | eqtri 2844 | . . . . . 6 ⊢ ({3} ∩ {1, 8}) = ∅ |
18 | disj3 4402 | . . . . . 6 ⊢ (({3} ∩ {1, 8}) = ∅ ↔ {3} = ({3} ∖ {1, 8})) | |
19 | 17, 18 | mpbi 232 | . . . . 5 ⊢ {3} = ({3} ∖ {1, 8}) |
20 | 19 | eqcomi 2830 | . . . 4 ⊢ ({3} ∖ {1, 8}) = {3} |
21 | 6, 20 | uneq12i 4136 | . . 3 ⊢ (({1} ∖ {1, 8}) ∪ ({3} ∖ {1, 8})) = (∅ ∪ {3}) |
22 | uncom 4128 | . . 3 ⊢ (∅ ∪ {3}) = ({3} ∪ ∅) | |
23 | un0 4343 | . . 3 ⊢ ({3} ∪ ∅) = {3} | |
24 | 21, 22, 23 | 3eqtri 2848 | . 2 ⊢ (({1} ∖ {1, 8}) ∪ ({3} ∖ {1, 8})) = {3} |
25 | 2, 3, 24 | 3eqtri 2848 | 1 ⊢ ({1, 3} ∖ {1, 8}) = {3} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2110 ∖ cdif 3932 ∪ cun 3933 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 {csn 4560 {cpr 4562 1c1 10532 3c3 11687 8c8 11692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |