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| Description: Example for df-dif 3954. 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 4629 | . . 3 ⊢ {1, 3} = ({1} ∪ {3}) | |
| 2 | 1 | difeq1i 4122 | . 2 ⊢ ({1, 3} ∖ {1, 8}) = (({1} ∪ {3}) ∖ {1, 8}) | 
| 3 | difundir 4291 | . 2 ⊢ (({1} ∪ {3}) ∖ {1, 8}) = (({1} ∖ {1, 8}) ∪ ({3} ∖ {1, 8})) | |
| 4 | snsspr1 4814 | . . . . 5 ⊢ {1} ⊆ {1, 8} | |
| 5 | ssdif0 4366 | . . . . 5 ⊢ ({1} ⊆ {1, 8} ↔ ({1} ∖ {1, 8}) = ∅) | |
| 6 | 4, 5 | mpbi 230 | . . . 4 ⊢ ({1} ∖ {1, 8}) = ∅ | 
| 7 | incom 4209 | . . . . . . 7 ⊢ ({3} ∩ {1, 8}) = ({1, 8} ∩ {3}) | |
| 8 | 1re 11261 | . . . . . . . . . 10 ⊢ 1 ∈ ℝ | |
| 9 | 1lt3 12439 | . . . . . . . . . 10 ⊢ 1 < 3 | |
| 10 | 8, 9 | gtneii 11373 | . . . . . . . . 9 ⊢ 3 ≠ 1 | 
| 11 | 3re 12346 | . . . . . . . . . 10 ⊢ 3 ∈ ℝ | |
| 12 | 3lt8 12462 | . . . . . . . . . 10 ⊢ 3 < 8 | |
| 13 | 11, 12 | ltneii 11374 | . . . . . . . . 9 ⊢ 3 ≠ 8 | 
| 14 | 10, 13 | nelpri 4655 | . . . . . . . 8 ⊢ ¬ 3 ∈ {1, 8} | 
| 15 | disjsn 4711 | . . . . . . . 8 ⊢ (({1, 8} ∩ {3}) = ∅ ↔ ¬ 3 ∈ {1, 8}) | |
| 16 | 14, 15 | mpbir 231 | . . . . . . 7 ⊢ ({1, 8} ∩ {3}) = ∅ | 
| 17 | 7, 16 | eqtri 2765 | . . . . . 6 ⊢ ({3} ∩ {1, 8}) = ∅ | 
| 18 | disj3 4454 | . . . . . 6 ⊢ (({3} ∩ {1, 8}) = ∅ ↔ {3} = ({3} ∖ {1, 8})) | |
| 19 | 17, 18 | mpbi 230 | . . . . 5 ⊢ {3} = ({3} ∖ {1, 8}) | 
| 20 | 19 | eqcomi 2746 | . . . 4 ⊢ ({3} ∖ {1, 8}) = {3} | 
| 21 | 6, 20 | uneq12i 4166 | . . 3 ⊢ (({1} ∖ {1, 8}) ∪ ({3} ∖ {1, 8})) = (∅ ∪ {3}) | 
| 22 | uncom 4158 | . . 3 ⊢ (∅ ∪ {3}) = ({3} ∪ ∅) | |
| 23 | un0 4394 | . . 3 ⊢ ({3} ∪ ∅) = {3} | |
| 24 | 21, 22, 23 | 3eqtri 2769 | . 2 ⊢ (({1} ∖ {1, 8}) ∪ ({3} ∖ {1, 8})) = {3} | 
| 25 | 2, 3, 24 | 3eqtri 2769 | 1 ⊢ ({1, 3} ∖ {1, 8}) = {3} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ∪ cun 3949 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 {csn 4626 {cpr 4628 1c1 11156 3c3 12322 8c8 12327 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 | 
| This theorem is referenced by: (None) | 
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