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Mirrors > Home > MPE Home > Th. List > grothpwex | Structured version Visualization version GIF version |
Description: Derive the Axiom of Power Sets from the Tarski-Grothendieck axiom ax-groth 10526. Note that ax-pow 5288 is not used by the proof. Use axpweq 5287 to obtain ax-pow 5288. Use pwex 5303 or pwexg 5301 instead. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grothpwex | ⊢ 𝒫 𝑥 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . . . . . . 7 ⊢ ((𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) → 𝒫 𝑧 ⊆ 𝑦) | |
2 | 1 | ralimi 3085 | . . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) → ∀𝑧 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑦) |
3 | pweq 4551 | . . . . . . . 8 ⊢ (𝑧 = 𝑥 → 𝒫 𝑧 = 𝒫 𝑥) | |
4 | 3 | sseq1d 3953 | . . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝒫 𝑧 ⊆ 𝑦 ↔ 𝒫 𝑥 ⊆ 𝑦)) |
5 | 4 | rspccv 3554 | . . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑦 → (𝑥 ∈ 𝑦 → 𝒫 𝑥 ⊆ 𝑦)) |
6 | 2, 5 | syl 17 | . . . . 5 ⊢ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) → (𝑥 ∈ 𝑦 → 𝒫 𝑥 ⊆ 𝑦)) |
7 | 6 | anim2i 616 | . . . 4 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤)) → (𝑥 ∈ 𝑦 ∧ (𝑥 ∈ 𝑦 → 𝒫 𝑥 ⊆ 𝑦))) |
8 | 7 | 3adant3 1130 | . . 3 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)) → (𝑥 ∈ 𝑦 ∧ (𝑥 ∈ 𝑦 → 𝒫 𝑥 ⊆ 𝑦))) |
9 | pm3.35 799 | . . 3 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑥 ∈ 𝑦 → 𝒫 𝑥 ⊆ 𝑦)) → 𝒫 𝑥 ⊆ 𝑦) | |
10 | vex 3431 | . . . 4 ⊢ 𝑦 ∈ V | |
11 | 10 | ssex 5245 | . . 3 ⊢ (𝒫 𝑥 ⊆ 𝑦 → 𝒫 𝑥 ∈ V) |
12 | 8, 9, 11 | 3syl 18 | . 2 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)) → 𝒫 𝑥 ∈ V) |
13 | axgroth5 10527 | . 2 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)) | |
14 | 12, 13 | exlimiiv 1935 | 1 ⊢ 𝒫 𝑥 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 843 ∧ w3a 1085 ∈ wcel 2107 ∀wral 3062 ∃wrex 3063 Vcvv 3427 ⊆ wss 3888 𝒫 cpw 4535 class class class wbr 5075 ≈ cen 8693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5223 ax-groth 10526 |
This theorem depends on definitions: df-bi 206 df-an 396 df-3an 1087 df-tru 1542 df-ex 1784 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3429 df-in 3895 df-ss 3905 df-pw 4537 |
This theorem is referenced by: (None) |
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