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Mirrors > Home > NFE Home > Th. List > unsneqsn | GIF version |
Description: If union with a singleton yields a singleton, then the first argument is either also the singleton or is the empty set. (Contributed by SF, 15-Jan-2015.) |
Ref | Expression |
---|---|
unsneqsn.1 | ⊢ B ∈ V |
Ref | Expression |
---|---|
unsneqsn | ⊢ ((A ∪ {B}) = {C} → (A = ∅ ∨ A = {B})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun2 3427 | . . . . . . 7 ⊢ {B} ⊆ (A ∪ {B}) | |
2 | unsneqsn.1 | . . . . . . . 8 ⊢ B ∈ V | |
3 | 2 | snid 3760 | . . . . . . 7 ⊢ B ∈ {B} |
4 | 1, 3 | sselii 3270 | . . . . . 6 ⊢ B ∈ (A ∪ {B}) |
5 | eleq2 2414 | . . . . . 6 ⊢ ((A ∪ {B}) = {C} → (B ∈ (A ∪ {B}) ↔ B ∈ {C})) | |
6 | 4, 5 | mpbii 202 | . . . . 5 ⊢ ((A ∪ {B}) = {C} → B ∈ {C}) |
7 | elsni 3757 | . . . . 5 ⊢ (B ∈ {C} → B = C) | |
8 | 6, 7 | syl 15 | . . . 4 ⊢ ((A ∪ {B}) = {C} → B = C) |
9 | sneq 3744 | . . . . . 6 ⊢ (B = C → {B} = {C}) | |
10 | 9 | eqeq2d 2364 | . . . . 5 ⊢ (B = C → ((A ∪ {B}) = {B} ↔ (A ∪ {B}) = {C})) |
11 | 10 | biimprd 214 | . . . 4 ⊢ (B = C → ((A ∪ {B}) = {C} → (A ∪ {B}) = {B})) |
12 | 8, 11 | mpcom 32 | . . 3 ⊢ ((A ∪ {B}) = {C} → (A ∪ {B}) = {B}) |
13 | ssequn1 3433 | . . 3 ⊢ (A ⊆ {B} ↔ (A ∪ {B}) = {B}) | |
14 | 12, 13 | sylibr 203 | . 2 ⊢ ((A ∪ {B}) = {C} → A ⊆ {B}) |
15 | sssn 3864 | . 2 ⊢ (A ⊆ {B} ↔ (A = ∅ ∨ A = {B})) | |
16 | 14, 15 | sylib 188 | 1 ⊢ ((A ∪ {B}) = {C} → (A = ∅ ∨ A = {B})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 357 = wceq 1642 ∈ wcel 1710 Vcvv 2859 ∪ cun 3207 ⊆ wss 3257 ∅c0 3550 {csn 3737 |
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-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-sn 3741 |
This theorem is referenced by: nnsucelr 4428 |
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