Theorem intssunim 3658

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 3329	. . . 4 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
2	1	ex 113	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥))
3		vex 2604	. . . 4 ⊢ 𝑦 ∈ V
4	3	elint2 3643	. . 3 ⊢ (𝑦 ∈ ∩ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
5		eluni2 3605	. . 3 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
6	2, 4, 5	3imtr4g 203	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑦 ∈ ∩ 𝐴 → 𝑦 ∈ ∪ 𝐴))
7	6	ssrdv 3005	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ ∪ 𝐴)

Syntax hints:   → wi 4  ∃wex 1421   ∈ wcel 1433  ∀wral 2348  ∃wrex 2349   ⊆ wss 2973  ∪ cuni 3601  ∩ cint 3636

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-in 2979  df-ss 2986  df-uni 3602  df-int 3637

This theorem is referenced by:  intssuni2m  3660