Theorem ssiun 3699

Description: Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
ssiun	⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Distinct variable group:   𝑥,𝐶

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem ssiun

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 2939	. . . . 5 ⊢ (𝐶 ⊆ 𝐵 → (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐵))
2	1	reximi 2416	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐵))
3		r19.37av 2463	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐵) → (𝑦 ∈ 𝐶 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
4	2, 3	syl 14	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → (𝑦 ∈ 𝐶 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
5		eliun 3661	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
6	4, 5	syl6ibr 151	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → (𝑦 ∈ 𝐶 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
7	6	ssrdv 2951	1 ⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   ∈ wcel 1393  ∃wrex 2307   ⊆ wss 2917  ∪ ciun 3657

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-iun 3659

This theorem is referenced by:  iunss2  3702  iunpwss  3743  iunpw  4211