Theorem ssiun 3930

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 3151	. . . . 5 ⊢ (𝐶 ⊆ 𝐵 → (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐵))
2	1	reximi 2574	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐵))
3		r19.37av 2630	. . . 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 3892	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
6	4, 5	imbitrrdi 162	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → (𝑦 ∈ 𝐶 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
7	6	ssrdv 3163	1 ⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2148  ∃wrex 2456   ⊆ wss 3131  ∪ ciun 3888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-in 3137  df-ss 3144  df-iun 3890

This theorem is referenced by:  iunss2  3933  iunpwss  3980  iunpw  4482