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Mirrors > Home > ILE Home > Th. List > snon0 | GIF version |
Description: An ordinal which is a singleton is {∅}. (Contributed by Jim Kingdon, 19-Oct-2021.) |
Ref | Expression |
---|---|
snon0 | ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elirr 4284 | . . 3 ⊢ ¬ 𝐴 ∈ 𝐴 | |
2 | snidg 3423 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴}) | |
3 | 2 | adantr 270 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → 𝐴 ∈ {𝐴}) |
4 | ontr1 4144 | . . . . . . 7 ⊢ ({𝐴} ∈ On → ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ {𝐴}) → 𝑥 ∈ {𝐴})) | |
5 | 4 | adantl 271 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ {𝐴}) → 𝑥 ∈ {𝐴})) |
6 | 3, 5 | mpan2d 418 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝐴})) |
7 | elsni 3416 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
8 | 6, 7 | syl6 33 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → (𝑥 ∈ 𝐴 → 𝑥 = 𝐴)) |
9 | eleq1 2141 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐴 ∈ 𝐴)) | |
10 | 9 | biimpcd 157 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = 𝐴 → 𝐴 ∈ 𝐴)) |
11 | 8, 10 | sylcom 28 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → (𝑥 ∈ 𝐴 → 𝐴 ∈ 𝐴)) |
12 | 1, 11 | mtoi 622 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → ¬ 𝑥 ∈ 𝐴) |
13 | 12 | eq0rdv 3288 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 ∅c0 3251 {csn 3398 Oncon0 4118 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-setind 4280 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-in 2979 df-ss 2986 df-nul 3252 df-sn 3404 df-uni 3602 df-tr 3876 df-iord 4121 df-on 4123 |
This theorem is referenced by: (None) |
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