Theorem zfcndpow 10685

Description: Axiom of Power Sets ax-pow 5383, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness" dtru 5456. Usage of this theorem is discouraged because it depends on ax-13 2380. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.) (New usage 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 5456	. . . . 5 ⊢ ¬ ∀𝑦 𝑦 = 𝑧
2		exnal 1825	. . . . 5 ⊢ (∃𝑦 ¬ 𝑦 = 𝑧 ↔ ¬ ∀𝑦 𝑦 = 𝑧)
3	1, 2	mpbir 231	. . . 4 ⊢ ∃𝑦 ¬ 𝑦 = 𝑧
4		nfe1 2151	. . . . 5 ⊢ Ⅎ𝑦∃𝑦∀𝑧(∀𝑦(∃𝑥 𝑦 ∈ 𝑧 → ∀𝑧 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦)
5		axpownd 10670	. . . . 5 ⊢ (¬ 𝑦 = 𝑧 → ∃𝑦∀𝑧(∀𝑦(∃𝑥 𝑦 ∈ 𝑧 → ∀𝑧 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
6	4, 5	exlimi 2218	. . . 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 1983	. . . . . . . 8 ⊢ (∃𝑥 𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)
9		19.3v 1981	. . . . . . . 8 ⊢ (∀𝑧 𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥)
10	8, 9	imbi12i 350	. . . . . . 7 ⊢ ((∃𝑥 𝑦 ∈ 𝑧 → ∀𝑧 𝑦 ∈ 𝑥) ↔ (𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥))
11	10	albii 1817	. . . . . 6 ⊢ (∀𝑦(∃𝑥 𝑦 ∈ 𝑧 → ∀𝑧 𝑦 ∈ 𝑥) ↔ ∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥))
12	11	imbi1i 349	. . . . 5 ⊢ ((∀𝑦(∃𝑥 𝑦 ∈ 𝑧 → ∀𝑧 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ (∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
13	12	albii 1817	. . . 4 ⊢ (∀𝑧(∀𝑦(∃𝑥 𝑦 ∈ 𝑧 → ∀𝑧 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ ∀𝑧(∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
14	13	exbii 1846	. . 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 230	. 2 ⊢ ∃𝑦∀𝑧(∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦)
16		elequ1 2115	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
17		elequ1 2115	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
18	16, 17	imbi12d 344	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) ↔ (𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥)))
19	18	cbvalvw 2035	. . . . 5 ⊢ (∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) ↔ ∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥))
20	19	imbi1i 349	. . . 4 ⊢ ((∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ (∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
21	20	albii 1817	. . 3 ⊢ (∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ ∀𝑧(∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
22	21	exbii 1846	. 2 ⊢ (∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ ∃𝑦∀𝑧(∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
23	15, 22	mpbir 231	1 ⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1535   = wceq 1537  ∃wex 1777   ∈ wcel 2108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-13 2380  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-reg 9661

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-pw 4624  df-sn 4649  df-pr 4651

This theorem is referenced by: (None)