Theorem iunlub 48693

Description: The indexed union is the the lowest upper bound if it exists. (Contributed by Zhi Wang, 1-Nov-2025.)

Hypotheses

Ref	Expression
iunlub.1	⊢ (𝜑 → 𝑋 ∈ 𝐴)
iunlub.2	⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝐵 = 𝐶)
iunlub.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)

Assertion

Ref	Expression
iunlub	⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝑋   𝜑,𝑥

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iunlub

Step	Hyp	Ref	Expression
1		iunlub.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)
2	1	iunssd 5024	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
3		iunlub.1	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
4		iunlub.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝐵 = 𝐶)
5	4	sseq2d 3989	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐶 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐶))
6		ssidd 3980	. . . 4 ⊢ (𝜑 → 𝐶 ⊆ 𝐶)
7	3, 5, 6	rspcedvd 3601	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)
8		ssiun 5020	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
9	7, 8	syl 17	. 2 ⊢ (𝜑 → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
10	2, 9	eqssd 3974	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∃wrex 3059   ⊆ wss 3924  ∪ ciun 4965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-v 3459  df-ss 3941  df-iun 4967

This theorem is referenced by: (None)