Metamath Proof Explorer


Theorem zfcndpow

Description: Axiom of Power Sets ax-pow , reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness" dtru . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 15-Aug-2003) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion zfcndpow 𝑦𝑧 ( ∀ 𝑤 ( 𝑤𝑧𝑤𝑥 ) → 𝑧𝑦 )

Proof

Step Hyp Ref Expression
1 dtru ¬ ∀ 𝑦 𝑦 = 𝑧
2 exnal ( ∃ 𝑦 ¬ 𝑦 = 𝑧 ↔ ¬ ∀ 𝑦 𝑦 = 𝑧 )
3 1 2 mpbir 𝑦 ¬ 𝑦 = 𝑧
4 nfe1 𝑦𝑦𝑧 ( ∀ 𝑦 ( ∃ 𝑥 𝑦𝑧 → ∀ 𝑧 𝑦𝑥 ) → 𝑧𝑦 )
5 axpownd ( ¬ 𝑦 = 𝑧 → ∃ 𝑦𝑧 ( ∀ 𝑦 ( ∃ 𝑥 𝑦𝑧 → ∀ 𝑧 𝑦𝑥 ) → 𝑧𝑦 ) )
6 4 5 exlimi ( ∃ 𝑦 ¬ 𝑦 = 𝑧 → ∃ 𝑦𝑧 ( ∀ 𝑦 ( ∃ 𝑥 𝑦𝑧 → ∀ 𝑧 𝑦𝑥 ) → 𝑧𝑦 ) )
7 3 6 ax-mp 𝑦𝑧 ( ∀ 𝑦 ( ∃ 𝑥 𝑦𝑧 → ∀ 𝑧 𝑦𝑥 ) → 𝑧𝑦 )
8 19.9v ( ∃ 𝑥 𝑦𝑧𝑦𝑧 )
9 19.3v ( ∀ 𝑧 𝑦𝑥𝑦𝑥 )
10 8 9 imbi12i ( ( ∃ 𝑥 𝑦𝑧 → ∀ 𝑧 𝑦𝑥 ) ↔ ( 𝑦𝑧𝑦𝑥 ) )
11 10 albii ( ∀ 𝑦 ( ∃ 𝑥 𝑦𝑧 → ∀ 𝑧 𝑦𝑥 ) ↔ ∀ 𝑦 ( 𝑦𝑧𝑦𝑥 ) )
12 11 imbi1i ( ( ∀ 𝑦 ( ∃ 𝑥 𝑦𝑧 → ∀ 𝑧 𝑦𝑥 ) → 𝑧𝑦 ) ↔ ( ∀ 𝑦 ( 𝑦𝑧𝑦𝑥 ) → 𝑧𝑦 ) )
13 12 albii ( ∀ 𝑧 ( ∀ 𝑦 ( ∃ 𝑥 𝑦𝑧 → ∀ 𝑧 𝑦𝑥 ) → 𝑧𝑦 ) ↔ ∀ 𝑧 ( ∀ 𝑦 ( 𝑦𝑧𝑦𝑥 ) → 𝑧𝑦 ) )
14 13 exbii ( ∃ 𝑦𝑧 ( ∀ 𝑦 ( ∃ 𝑥 𝑦𝑧 → ∀ 𝑧 𝑦𝑥 ) → 𝑧𝑦 ) ↔ ∃ 𝑦𝑧 ( ∀ 𝑦 ( 𝑦𝑧𝑦𝑥 ) → 𝑧𝑦 ) )
15 7 14 mpbi 𝑦𝑧 ( ∀ 𝑦 ( 𝑦𝑧𝑦𝑥 ) → 𝑧𝑦 )
16 elequ1 ( 𝑤 = 𝑦 → ( 𝑤𝑧𝑦𝑧 ) )
17 elequ1 ( 𝑤 = 𝑦 → ( 𝑤𝑥𝑦𝑥 ) )
18 16 17 imbi12d ( 𝑤 = 𝑦 → ( ( 𝑤𝑧𝑤𝑥 ) ↔ ( 𝑦𝑧𝑦𝑥 ) ) )
19 18 cbvalvw ( ∀ 𝑤 ( 𝑤𝑧𝑤𝑥 ) ↔ ∀ 𝑦 ( 𝑦𝑧𝑦𝑥 ) )
20 19 imbi1i ( ( ∀ 𝑤 ( 𝑤𝑧𝑤𝑥 ) → 𝑧𝑦 ) ↔ ( ∀ 𝑦 ( 𝑦𝑧𝑦𝑥 ) → 𝑧𝑦 ) )
21 20 albii ( ∀ 𝑧 ( ∀ 𝑤 ( 𝑤𝑧𝑤𝑥 ) → 𝑧𝑦 ) ↔ ∀ 𝑧 ( ∀ 𝑦 ( 𝑦𝑧𝑦𝑥 ) → 𝑧𝑦 ) )
22 21 exbii ( ∃ 𝑦𝑧 ( ∀ 𝑤 ( 𝑤𝑧𝑤𝑥 ) → 𝑧𝑦 ) ↔ ∃ 𝑦𝑧 ( ∀ 𝑦 ( 𝑦𝑧𝑦𝑥 ) → 𝑧𝑦 ) )
23 15 22 mpbir 𝑦𝑧 ( ∀ 𝑤 ( 𝑤𝑧𝑤𝑥 ) → 𝑧𝑦 )