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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrclsrcomplex | Structured version Visualization version GIF version |
Description: The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 25-Jun-2021.) |
Ref | Expression |
---|---|
ntrclsbex.d | ⊢ 𝐷 = (𝑂‘𝐵) |
ntrclsbex.r | ⊢ (𝜑 → 𝐼𝐷𝐾) |
Ref | Expression |
---|---|
ntrclsrcomplex | ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrclsbex.d | . . 3 ⊢ 𝐷 = (𝑂‘𝐵) | |
2 | ntrclsbex.r | . . 3 ⊢ (𝜑 → 𝐼𝐷𝐾) | |
3 | 1, 2 | ntrclsbex 40391 | . 2 ⊢ (𝜑 → 𝐵 ∈ V) |
4 | difssd 4111 | . 2 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ⊆ 𝐵) | |
5 | 3, 4 | sselpwd 5232 | 1 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∖ cdif 3935 𝒫 cpw 4541 class class class wbr 5068 ‘cfv 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 |
This theorem is referenced by: ntrclsfveq1 40417 ntrclsfveq2 40418 ntrclsfveq 40419 ntrclsss 40420 ntrclsneine0lem 40421 ntrclsk2 40425 ntrclskb 40426 ntrclsk4 40429 |
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