Theorem zfcndinf 10706

Description: Axiom of Infinity ax-inf 9642, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing Theorem el 5405 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 5405	. . 3 ⊢ ∃𝑤 𝑥 ∈ 𝑤
2		nfv 1946	. . . . . 6 ⊢ Ⅎ𝑤 𝑥 ∈ 𝑦
3		nfe1 2186	. . . . . . . 8 ⊢ Ⅎ𝑤∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)
4	2, 3	nfim 1928	. . . . . . 7 ⊢ Ⅎ𝑤(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))
5	4	nfal 2353	. . . . . 6 ⊢ Ⅎ𝑤∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))
6	2, 5	nfan 1931	. . . . 5 ⊢ Ⅎ𝑤(𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
7	6	nfex 2354	. . . 4 ⊢ Ⅎ𝑤∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
8		axinfnd 10694	. . . . 5 ⊢ ∃𝑦(𝑥 ∈ 𝑤 → (𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))
9	8	19.37iv 1980	. . . 4 ⊢ (𝑥 ∈ 𝑤 → ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))
10	7, 9	exlimi 2253	. . 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 2151	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
13		elequ1 2151	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑤 ↔ 𝑥 ∈ 𝑤))
14	13	anbi1d 643	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦) ↔ (𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
15	14	exbidv 1953	. . . . . 6 ⊢ (𝑧 = 𝑥 → (∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦) ↔ ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
16	12, 15	imbi12d 347	. . . . 5 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)) ↔ (𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))
17	16	cbvalvw 2068	. . . 4 ⊢ (∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)) ↔ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
18	17	anbi2i 635	. . 3 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))) ↔ (𝑥 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑦 → ∃𝑤(𝑥 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))))
19	18	exbii 1880	. 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 234	1 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-13 2401  ax-ext 2732  ax-sep 5248  ax-pr 5390  ax-reg 9589  ax-inf 9642

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-cleq 2752  df-clel 2835  df-nfc 2909

This theorem is used by: (None)