Theorem iunssfOLD 5002

Description: Obsolete version of iunssf 5001 as of 2-Feb-2026. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
iunssf.1	⊢ Ⅎ𝑥𝐶

Assertion

Ref	Expression
iunssfOLD	⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)

Proof of Theorem iunssfOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 4953	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
2	1	sseq1i 3959	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ⊆ 𝐶)
3		abss 4010	. 2 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ⊆ 𝐶 ↔ ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
4		df-ss 3916	. . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
5	4	ralbii 3109	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
6		ralcom4 3289	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
7		iunssf.1	. . . . . 6 ⊢ Ⅎ𝑥𝐶
8	7	nfcri 2915	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐶
9	8	r19.23 3260	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
10	9	albii 1852	. . 3 ⊢ (∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) ↔ ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
11	5, 6, 10	3bitrri 301	. 2 ⊢ (∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
12	2, 3, 11	3bitri 300	1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   ∈ wcel 2145  {cab 2739  Ⅎwnfc 2908  ∀wral 3077  ∃wrex 3087   ⊆ wss 3899  ∪ ciun 4951

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-ss 3916  df-iun 4953

This theorem is used by: (None)