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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 10247. Note that ax-pow 5268 is not used by the proof. Use axpweq 5267 to obtain ax-pow 5268. Use pwex 5283 or pwexg 5281 instead. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grothpwex | ⊢ 𝒫 𝑥 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 485 | . . . . . . 7 ⊢ ((𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) → 𝒫 𝑧 ⊆ 𝑦) | |
2 | 1 | ralimi 3162 | . . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) → ∀𝑧 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑦) |
3 | pweq 4557 | . . . . . . . 8 ⊢ (𝑧 = 𝑥 → 𝒫 𝑧 = 𝒫 𝑥) | |
4 | 3 | sseq1d 4000 | . . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝒫 𝑧 ⊆ 𝑦 ↔ 𝒫 𝑥 ⊆ 𝑦)) |
5 | 4 | rspccv 3622 | . . . . . 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 618 | . . . 4 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤)) → (𝑥 ∈ 𝑦 ∧ (𝑥 ∈ 𝑦 → 𝒫 𝑥 ⊆ 𝑦))) |
8 | 7 | 3adant3 1128 | . . 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 | pm3.35 801 | . . 3 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑥 ∈ 𝑦 → 𝒫 𝑥 ⊆ 𝑦)) → 𝒫 𝑥 ⊆ 𝑦) | |
10 | vex 3499 | . . . 4 ⊢ 𝑦 ∈ V | |
11 | 10 | ssex 5227 | . . 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 10248 | . 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 1932 | 1 ⊢ 𝒫 𝑥 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 Vcvv 3496 ⊆ wss 3938 𝒫 cpw 4541 class class class wbr 5068 ≈ cen 8508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-groth 10247 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-in 3945 df-ss 3954 df-pw 4543 |
This theorem is referenced by: (None) |
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