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Mirrors > Home > MPE Home > Th. List > rabexd | Structured version Visualization version GIF version |
Description: Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 4742. (Contributed by AV, 16-Jul-2019.) |
Ref | Expression |
---|---|
rabexd.1 | ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓} |
rabexd.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
rabexd | ⊢ (𝜑 → 𝐵 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabexd.1 | . 2 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓} | |
2 | rabexd.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | rabexg 4739 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V) |
5 | 1, 4 | syl5eqel 2692 | 1 ⊢ (𝜑 → 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 |
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-rab 2905 df-v 3175 df-in 3547 df-ss 3554 |
This theorem is referenced by: rabex2 4742 rabex2OLD 4744 zorn2lem1 9201 sylow2a 17857 evlslem6 19334 mretopd 20706 cusgraexilem1 25995 stoweidlem35 38928 stoweidlem50 38943 stoweidlem57 38950 stoweidlem59 38952 subsaliuncllem 39251 subsaliuncl 39252 smflimlem1 39657 smflimlem2 39658 smflimlem3 39659 smflimlem6 39662 smfrec 39674 usgrexi 40661 vtxdgf 40686 |
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