Theorem zfcndpow 9836

Description: Axiom of Power Sets ax-pow 5119, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness" dtru 5124. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)

Assertion

Ref	Expression
zfcndpow	⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Proof of Theorem zfcndpow

Step	Hyp	Ref	Expression
1		dtru 5124	. . . . 5 ⊢ ¬ ∀𝑦 𝑦 = 𝑧
2		exnal 1789	. . . . 5 ⊢ (∃𝑦 ¬ 𝑦 = 𝑧 ↔ ¬ ∀𝑦 𝑦 = 𝑧)
3	1, 2	mpbir 223	. . . 4 ⊢ ∃𝑦 ¬ 𝑦 = 𝑧
4		nfe1 2087	. . . . 5 ⊢ Ⅎ𝑦∃𝑦∀𝑧(∀𝑦(∃𝑥 𝑦 ∈ 𝑧 → ∀𝑧 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦)
5		axpownd 9821	. . . . 5 ⊢ (¬ 𝑦 = 𝑧 → ∃𝑦∀𝑧(∀𝑦(∃𝑥 𝑦 ∈ 𝑧 → ∀𝑧 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
6	4, 5	exlimi 2147	. . . 4 ⊢ (∃𝑦 ¬ 𝑦 = 𝑧 → ∃𝑦∀𝑧(∀𝑦(∃𝑥 𝑦 ∈ 𝑧 → ∀𝑧 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
7	3, 6	ax-mp 5	. . 3 ⊢ ∃𝑦∀𝑧(∀𝑦(∃𝑥 𝑦 ∈ 𝑧 → ∀𝑧 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦)
8		19.9v 1938	. . . . . . . 8 ⊢ (∃𝑥 𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)
9		19.3v 1939	. . . . . . . 8 ⊢ (∀𝑧 𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥)
10	8, 9	imbi12i 343	. . . . . . 7 ⊢ ((∃𝑥 𝑦 ∈ 𝑧 → ∀𝑧 𝑦 ∈ 𝑥) ↔ (𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥))
11	10	albii 1782	. . . . . 6 ⊢ (∀𝑦(∃𝑥 𝑦 ∈ 𝑧 → ∀𝑧 𝑦 ∈ 𝑥) ↔ ∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥))
12	11	imbi1i 342	. . . . 5 ⊢ ((∀𝑦(∃𝑥 𝑦 ∈ 𝑧 → ∀𝑧 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ (∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
13	12	albii 1782	. . . 4 ⊢ (∀𝑧(∀𝑦(∃𝑥 𝑦 ∈ 𝑧 → ∀𝑧 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ ∀𝑧(∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
14	13	exbii 1810	. . 3 ⊢ (∃𝑦∀𝑧(∀𝑦(∃𝑥 𝑦 ∈ 𝑧 → ∀𝑧 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ ∃𝑦∀𝑧(∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
15	7, 14	mpbi 222	. 2 ⊢ ∃𝑦∀𝑧(∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦)
16		elequ1 2057	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
17		elequ1 2057	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
18	16, 17	imbi12d 337	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) ↔ (𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥)))
19	18	cbvalvw 1993	. . . . 5 ⊢ (∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) ↔ ∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥))
20	19	imbi1i 342	. . . 4 ⊢ ((∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ (∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
21	20	albii 1782	. . 3 ⊢ (∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ ∀𝑧(∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
22	21	exbii 1810	. 2 ⊢ (∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ ∃𝑦∀𝑧(∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
23	15, 22	mpbir 223	1 ⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1505   = wceq 1507  ∃wex 1742   ∈ wcel 2050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-reg 8851

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-v 3417  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-pw 4424  df-sn 4442  df-pr 4444

This theorem is referenced by: (None)