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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 5335. (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 5332 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V) |
5 | 1, 4 | eqeltrid 2838 | 1 ⊢ (𝜑 → 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {crab 3433 Vcvv 3475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-in 3956 df-ss 3966 |
This theorem is referenced by: rabex2 5335 zorn2lem1 10491 sylow2a 19487 evlslem6 21644 mhpaddcl 21694 mhpvscacl 21697 mretopd 22596 cusgrexilem1 28696 vtxdgf 28728 mntoval 32152 tocycval 32267 prmidlval 32555 evlsvvval 41135 evlsbagval 41138 mhpind 41166 stoweidlem35 44751 stoweidlem50 44766 stoweidlem57 44773 stoweidlem59 44775 subsaliuncllem 45073 subsaliuncl 45074 smflimlem1 45487 smflimlem2 45488 smflimlem3 45489 smflimlem6 45492 smfrec 45505 smfpimcclem 45523 smfsuplem1 45527 smfinflem 45533 smflimsuplem1 45536 smflimsuplem2 45537 smflimsuplem3 45538 smflimsuplem4 45539 smflimsuplem5 45540 smflimsuplem7 45542 fvmptrab 46000 prproropen 46176 |
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