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Mirrors > Home > MPE Home > Th. List > difexi | Structured version Visualization version GIF version |
Description: Existence of a difference, inference version of difexg 4735. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Revised by AV, 26-Mar-2021.) |
Ref | Expression |
---|---|
difexi.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
difexi | ⊢ (𝐴 ∖ 𝐵) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difexi.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | difexg 4735 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∖ 𝐵) ∈ V) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∖ 𝐵) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 |
This theorem is referenced by: marypha1lem 8222 inf3lem3 8410 setindtr 36609 clsk3nimkb 37358 meaiuninclem 39373 meaiininclem 39376 upgrres1lem1 40528 nbgrval 40560 |
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