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Mirrors > Home > NFE Home > Th. List > un0 | GIF version |
Description: The union of a class with the empty set is itself. Theorem 24 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
un0 | ⊢ (A ∪ ∅) = A |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3554 | . . . 4 ⊢ ¬ x ∈ ∅ | |
2 | 1 | biorfi 396 | . . 3 ⊢ (x ∈ A ↔ (x ∈ A ∨ x ∈ ∅)) |
3 | 2 | bicomi 193 | . 2 ⊢ ((x ∈ A ∨ x ∈ ∅) ↔ x ∈ A) |
4 | 3 | uneqri 3406 | 1 ⊢ (A ∪ ∅) = A |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 357 = wceq 1642 ∈ wcel 1710 ∪ cun 3207 ∅c0 3550 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-nul 3551 |
This theorem is referenced by: un00 3586 disjssun 3608 difun2 3629 difdifdir 3637 diftpsn3 3849 sspr 3869 sstp 3870 iununi 4050 prprc2 4122 addcid1 4405 nnsucelrlem3 4426 fvun1 5379 fvunsn 5444 fvsnun1 5447 fvsnun2 5448 sbthlem1 6203 |
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