Theorem zfcndinf 10506

Description: Axiom of Infinity ax-inf 9528, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing Theorem el 5380 in the proof. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by NM, 15-Aug-2003.)

Assertion

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

Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Proof of Theorem zfcndinf

Step	Hyp	Ref	Expression
1		el 5380	. . 3 ⊢ ∃𝑤 𝑥 ∈ 𝑤
2		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑤 𝑥 ∈ 𝑦
3		nfe1 2153	. . . . . . . 8 ⊢ Ⅎ𝑤∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)
4	2, 3	nfim 1897	. . . . . . 7 ⊢ Ⅎ𝑤(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))
5	4	nfal 2324	. . . . . 6 ⊢ Ⅎ𝑤∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))
6	2, 5	nfan 1900	. . . . 5 ⊢ Ⅎ𝑤(𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
7	6	nfex 2325	. . . 4 ⊢ Ⅎ𝑤∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
8		axinfnd 10494	. . . . 5 ⊢ ∃𝑦(𝑥 ∈ 𝑤 → (𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))
9	8	19.37iv 1949	. . . 4 ⊢ (𝑥 ∈ 𝑤 → ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))
10	7, 9	exlimi 2220	. . 3 ⊢ (∃𝑤 𝑥 ∈ 𝑤 → ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))
11	1, 10	ax-mp 5	. 2 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
12		elequ1 2118	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
13		elequ1 2118	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑤 ↔ 𝑥 ∈ 𝑤))
14	13	anbi1d 631	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦) ↔ (𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
15	14	exbidv 1922	. . . . . 6 ⊢ (𝑧 = 𝑥 → (∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦) ↔ ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
16	12, 15	imbi12d 344	. . . . 5 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)) ↔ (𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))
17	16	cbvalvw 2037	. . . 4 ⊢ (∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)) ↔ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
18	17	anbi2i 623	. . 3 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))) ↔ (𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))
19	18	exbii 1849	. 2 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))
20	11, 19	mpbir 231	1 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2111

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-13 2372  ax-ext 2703  ax-sep 5234  ax-pr 5370  ax-reg 9478  ax-inf 9528

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-cleq 2723  df-clel 2806  df-nfc 2881

This theorem is referenced by: (None)