Theorem elssuni 1940

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.

Assertion

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

Proof of Theorem elssuni

Step	Hyp	Ref	Expression
1		ssid 1519	. 2 ⊢ A ⊆ A
2		ssuni 1937	. 2 ⊢ ((A ⊆ A ∧ A ∈ B) → A ⊆ ∪B)
3	1, 2	mpan 518	1 ⊢ (A ∈ B → A ⊆ ∪B)

Syntax hints:   → wi 2   ∈ wcel 1092   ⊆ wss 1487  ∪cuni 1919

This theorem is referenced by:  ssunieq 1945  pwuni 1961  iunpw 2040  tfrlem8 2956  tfrlem9 2957  tfrlem13 2961  sbthlem1 3349  sbthlem2 3350  rankuni 3533  kmlem2 3581  carduni 3664  cardprc 3667  cardinfima 3696  suplem2pr 3956  hatomistic 5755

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-ss 1492  df-uni 1920

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