Theorem zfcndun 10194

Description: Axiom of Union ax-un 7501, reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Proof of Theorem zfcndun

Step	Hyp	Ref	Expression
1		axunnd 10175	. 2 ⊢ ∃𝑦∀𝑧(∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦)
2		elequ2 2127	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦))
3		elequ1 2119	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
4	2, 3	anbi12d 634	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥)))
5	4	cbvexvw 2047	. . . . 5 ⊢ (∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥))
6	5	imbi1i 353	. . . 4 ⊢ ((∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ (∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
7	6	albii 1827	. . 3 ⊢ (∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ ∀𝑧(∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
8	7	exbii 1855	. 2 ⊢ (∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ ∃𝑦∀𝑧(∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
9	1, 8	mpbir 234	1 ⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1541   = wceq 1543  ∃wex 1787   ∈ wcel 2112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-13 2371  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-un 7501  ax-reg 9186

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-br 5040  df-opab 5102  df-eprel 5445  df-fr 5494

This theorem is referenced by: (None)