Theorem axunndlem1

Description: Lemma for the Axiom of Union with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Jan-2002) (New usage is discouraged.)

	Ref	Expression
Assertion	axunndlem1	⊢ ∃ 𝑥 ∀ 𝑦 ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		en2lp	⊢ ¬ ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 )
2		elequ2	⊢ ( 𝑦 = 𝑧 → ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧 ) )
3	2	anbi2d	⊢ ( 𝑦 = 𝑧 → ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ↔ ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) ) )
4	1 3	mtbii	⊢ ( 𝑦 = 𝑧 → ¬ ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) )
5	4	sps	⊢ ( ∀ 𝑦 𝑦 = 𝑧 → ¬ ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) )
6	5	nexdv	⊢ ( ∀ 𝑦 𝑦 = 𝑧 → ¬ ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) )
7	6	pm2.21d	⊢ ( ∀ 𝑦 𝑦 = 𝑧 → ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) )
8	7	axc4i	⊢ ( ∀ 𝑦 𝑦 = 𝑧 → ∀ 𝑦 ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) )
9	8	19.8ad	⊢ ( ∀ 𝑦 𝑦 = 𝑧 → ∃ 𝑥 ∀ 𝑦 ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) )
10		zfun	⊢ ∃ 𝑥 ∀ 𝑤 ( ∃ 𝑥 ( 𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑤 ∈ 𝑥 )
11		nfnae	⊢ Ⅎ 𝑦 ¬ ∀ 𝑦 𝑦 = 𝑧
12		nfnae	⊢ Ⅎ 𝑥 ¬ ∀ 𝑦 𝑦 = 𝑧
13		nfvd	⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → Ⅎ 𝑦 𝑤 ∈ 𝑥 )
14		nfcvf	⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → Ⅎ 𝑦 𝑧 )
15	14	nfcrd	⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → Ⅎ 𝑦 𝑥 ∈ 𝑧 )
16	13 15	nfand	⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → Ⅎ 𝑦 ( 𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) )
17	12 16	nfexd	⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → Ⅎ 𝑦 ∃ 𝑥 ( 𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) )
18	17 13	nfimd	⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → Ⅎ 𝑦 ( ∃ 𝑥 ( 𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑤 ∈ 𝑥 ) )
19		elequ1	⊢ ( 𝑤 = 𝑦 → ( 𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥 ) )
20	19	anbi1d	⊢ ( 𝑤 = 𝑦 → ( ( 𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) ↔ ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) ) )
21	20	exbidv	⊢ ( 𝑤 = 𝑦 → ( ∃ 𝑥 ( 𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) ↔ ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) ) )
22	21 19	imbi12d	⊢ ( 𝑤 = 𝑦 → ( ( ∃ 𝑥 ( 𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑤 ∈ 𝑥 ) ↔ ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ) )
23	22	a1i	⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → ( 𝑤 = 𝑦 → ( ( ∃ 𝑥 ( 𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑤 ∈ 𝑥 ) ↔ ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ) ) )
24	11 18 23	cbvald	⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → ( ∀ 𝑤 ( ∃ 𝑥 ( 𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑤 ∈ 𝑥 ) ↔ ∀ 𝑦 ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ) )
25	24	exbidv	⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → ( ∃ 𝑥 ∀ 𝑤 ( ∃ 𝑥 ( 𝑤 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑤 ∈ 𝑥 ) ↔ ∃ 𝑥 ∀ 𝑦 ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ) )
26	10 25	mpbii	⊢ ( ¬ ∀ 𝑦 𝑦 = 𝑧 → ∃ 𝑥 ∀ 𝑦 ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) )
27	9 26	pm2.61i	⊢ ∃ 𝑥 ∀ 𝑦 ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 )