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Mirrors > Home > ILE 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 3408 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
2 | 1 | biorfi 736 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ ∅)) |
3 | 2 | bicomi 131 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ ∅) ↔ 𝑥 ∈ 𝐴) |
4 | 3 | uneqri 3259 | 1 ⊢ (𝐴 ∪ ∅) = 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∨ wo 698 = wceq 1342 ∈ wcel 2135 ∪ cun 3109 ∅c0 3404 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-dif 3113 df-un 3115 df-nul 3405 |
This theorem is referenced by: un00 3450 disjssun 3467 difun2 3483 difdifdirss 3488 disjpr2 3634 prprc1 3678 diftpsn3 3708 iununir 3943 suc0 4383 sucprc 4384 fvun1 5546 fmptpr 5671 fvunsng 5673 fvsnun1 5676 fvsnun2 5677 fsnunfv 5680 fsnunres 5681 rdg0 6346 omv2 6424 unsnfidcex 6876 unfidisj 6878 undifdc 6880 ssfirab 6890 dju0en 7161 djuassen 7164 fzsuc2 10004 fseq1p1m1 10019 hashunlem 10706 ennnfonelem1 12283 setsresg 12375 setsslid 12387 fmelpw1o 13529 exmid1stab 13721 |
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