Theorem ssiun2s 4011

Description: Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.)

Hypothesis

Ref	Expression
ssiun2s.1	⊢ (x = C → B = D)

Assertion

Ref	Expression
ssiun2s	⊢ (C ∈ A → D ⊆ ∪x ∈ A B)

Distinct variable groups:   x,A   x,C   x,D

Allowed substitution hint:   B(x)

Proof of Theorem ssiun2s

Step	Hyp	Ref	Expression
1		nfcv 2490	. 2 ⊢ ℲxC
2		nfcv 2490	. . 3 ⊢ ℲxD
3		nfiu1 3998	. . 3 ⊢ Ⅎx∪x ∈ A B
4	2, 3	nfss 3267	. 2 ⊢ Ⅎx D ⊆ ∪x ∈ A B
5		ssiun2s.1	. . 3 ⊢ (x = C → B = D)
6	5	sseq1d 3299	. 2 ⊢ (x = C → (B ⊆ ∪x ∈ A B ↔ D ⊆ ∪x ∈ A B))
7		ssiun2 4010	. 2 ⊢ (x ∈ A → B ⊆ ∪x ∈ A B)
8	1, 4, 6, 7	vtoclgaf 2920	1 ⊢ (C ∈ A → D ⊆ ∪x ∈ A B)

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710   ⊆ wss 3258  ∪ciun 3970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-iun 3972

This theorem is referenced by: (None)