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Mirrors > Home > MPE Home > Th. List > Mathboxes > exrecfnpw | Structured version Visualization version GIF version |
Description: For any base set, a set which contains the powerset of all of its own elements exists. (Contributed by ML, 30-Mar-2022.) |
Ref | Expression |
---|---|
exrecfnpw | ⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝒫 𝑦 ∈ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vpwex 5379 | . . 3 ⊢ 𝒫 𝑦 ∈ V | |
2 | 1 | ax-gen 1790 | . 2 ⊢ ∀𝑦𝒫 𝑦 ∈ V |
3 | exrecfn 37323 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦𝒫 𝑦 ∈ V) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝒫 𝑦 ∈ 𝑥)) | |
4 | 2, 3 | mpan2 690 | 1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝒫 𝑦 ∈ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1533 ∃wex 1774 ∈ wcel 2104 ∀wral 3057 Vcvv 3477 ⊆ wss 3963 𝒫 cpw 4605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7748 ax-inf2 9673 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4916 df-iun 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6318 df-ord 6384 df-on 6385 df-lim 6386 df-suc 6387 df-iota 6511 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-ov 7429 df-om 7882 df-2nd 8009 df-frecs 8300 df-wrecs 8331 df-recs 8405 df-rdg 8444 |
This theorem is referenced by: (None) |
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