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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 3426 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
2 | 1 | biorfi 746 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ ∅)) |
3 | 2 | bicomi 132 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ ∅) ↔ 𝑥 ∈ 𝐴) |
4 | 3 | uneqri 3277 | 1 ⊢ (𝐴 ∪ ∅) = 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∨ wo 708 = wceq 1353 ∈ wcel 2148 ∪ cun 3127 ∅c0 3422 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-dif 3131 df-un 3133 df-nul 3423 |
This theorem is referenced by: un00 3469 disjssun 3486 difun2 3502 difdifdirss 3507 disjpr2 3655 prprc1 3699 diftpsn3 3732 iununir 3967 exmid1stab 4205 suc0 4408 sucprc 4409 fvun1 5578 fmptpr 5704 fvunsng 5706 fvsnun1 5709 fvsnun2 5710 fsnunfv 5713 fsnunres 5714 rdg0 6382 omv2 6460 unsnfidcex 6913 unfidisj 6915 undifdc 6917 ssfirab 6927 dju0en 7207 djuassen 7210 fzsuc2 10062 fseq1p1m1 10077 hashunlem 10765 ennnfonelem1 12388 setsresg 12480 setsslid 12492 fmelpw1o 14207 |
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