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Mirrors > Home > MPE Home > Th. List > cnvin | Structured version Visualization version GIF version |
Description: Distributive law for converse over intersection. Theorem 15 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Revised by Mario Carneiro, 26-Jun-2014.) |
Ref | Expression |
---|---|
cnvin | ⊢ ◡(𝐴 ∩ 𝐵) = (◡𝐴 ∩ ◡𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvdif 6143 | . . 3 ⊢ ◡(𝐴 ∖ (𝐴 ∖ 𝐵)) = (◡𝐴 ∖ ◡(𝐴 ∖ 𝐵)) | |
2 | cnvdif 6143 | . . . 4 ⊢ ◡(𝐴 ∖ 𝐵) = (◡𝐴 ∖ ◡𝐵) | |
3 | 2 | difeq2i 4119 | . . 3 ⊢ (◡𝐴 ∖ ◡(𝐴 ∖ 𝐵)) = (◡𝐴 ∖ (◡𝐴 ∖ ◡𝐵)) |
4 | 1, 3 | eqtri 2760 | . 2 ⊢ ◡(𝐴 ∖ (𝐴 ∖ 𝐵)) = (◡𝐴 ∖ (◡𝐴 ∖ ◡𝐵)) |
5 | dfin4 4267 | . . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵)) | |
6 | 5 | cnveqi 5874 | . 2 ⊢ ◡(𝐴 ∩ 𝐵) = ◡(𝐴 ∖ (𝐴 ∖ 𝐵)) |
7 | dfin4 4267 | . 2 ⊢ (◡𝐴 ∩ ◡𝐵) = (◡𝐴 ∖ (◡𝐴 ∖ ◡𝐵)) | |
8 | 4, 6, 7 | 3eqtr4i 2770 | 1 ⊢ ◡(𝐴 ∩ 𝐵) = (◡𝐴 ∩ ◡𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∖ cdif 3945 ∩ cin 3947 ◡ccnv 5675 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-cnv 5684 |
This theorem is referenced by: rnin 6146 dminxp 6179 imainrect 6180 cnvcnv 6191 cnvrescnv 6194 pjdm 21261 ordtrest2 22707 ustexsym 23719 trust 23733 ordtcnvNEW 32895 ordtrest2NEW 32898 msrf 34528 elrn3 34727 pprodcnveq 34850 |
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