Theorem intssunim 3801

Description: The intersection of an inhabited set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)

Assertion

Ref	Expression
intssunim	⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ ∪ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem intssunim

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.2m 3454	. . . 4 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
2	1	ex 114	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥))
3		vex 2692	. . . 4 ⊢ 𝑦 ∈ V
4	3	elint2 3786	. . 3 ⊢ (𝑦 ∈ ∩ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
5		eluni2 3748	. . 3 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
6	2, 4, 5	3imtr4g 204	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑦 ∈ ∩ 𝐴 → 𝑦 ∈ ∪ 𝐴))
7	6	ssrdv 3108	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ ∪ 𝐴)

Syntax hints:   → wi 4  ∃wex 1469   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418   ⊆ wss 3076  ∪ cuni 3744  ∩ cint 3779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3082  df-ss 3089  df-uni 3745  df-int 3780

This theorem is referenced by:  intssuni2m  3803