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Mirrors > Home > NFE Home > Th. List > unisn | GIF version |
Description: A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 30-Aug-1993.) |
Ref | Expression |
---|---|
unisn.1 | ⊢ A ∈ V |
Ref | Expression |
---|---|
unisn | ⊢ ∪{A} = A |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsn2 3747 | . . 3 ⊢ {A} = {A, A} | |
2 | 1 | unieqi 3901 | . 2 ⊢ ∪{A} = ∪{A, A} |
3 | unisn.1 | . . 3 ⊢ A ∈ V | |
4 | 3, 3 | unipr 3905 | . 2 ⊢ ∪{A, A} = (A ∪ A) |
5 | unidm 3407 | . 2 ⊢ (A ∪ A) = A | |
6 | 2, 4, 5 | 3eqtri 2377 | 1 ⊢ ∪{A} = A |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ∈ wcel 1710 Vcvv 2859 ∪ cun 3207 {csn 3737 {cpr 3738 ∪cuni 3891 |
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-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-un 3214 df-sn 3741 df-pr 3742 df-uni 3892 |
This theorem is referenced by: unisng 3908 uniintsn 3963 pw1eqadj 4332 uniabio 4349 nnadjoin 4520 op1sta 5072 opswap 5074 op2nda 5076 funfv 5375 pw1fnval 5851 pw1fnf1o 5855 brtcfn 6246 |
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