Theorem elssuni 3920

Description: An element of a class is a subclass of its union. Theorem 8.6 of [Quine] p. 54. Also the basis for Proposition 7.20 of [TakeutiZaring] p. 40. (Contributed by NM, 6-Jun-1994.)

Assertion

Ref	Expression
elssuni	⊢ (A ∈ B → A ⊆ ∪B)

Proof of Theorem elssuni

Step	Hyp	Ref	Expression
1		ssid 3291	. 2 ⊢ A ⊆ A
2		ssuni 3914	. 2 ⊢ ((A ⊆ A ∧ A ∈ B) → A ⊆ ∪B)
3	1, 2	mpan 651	1 ⊢ (A ∈ B → A ⊆ ∪B)

Syntax hints:   → wi 4   ∈ wcel 1710   ⊆ wss 3258  ∪cuni 3892

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-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-uni 3893

This theorem is referenced by:  unissel  3921  ssunieq  3925  ncssfin  6152