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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 4464 | . . 3 ⊢ ¬ 𝐴 ∈ 𝐴 | |
2 | snidg 3561 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴}) | |
3 | 2 | adantr 274 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → 𝐴 ∈ {𝐴}) |
4 | ontr1 4319 | . . . . . . 7 ⊢ ({𝐴} ∈ On → ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ {𝐴}) → 𝑥 ∈ {𝐴})) | |
5 | 4 | adantl 275 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ {𝐴}) → 𝑥 ∈ {𝐴})) |
6 | 3, 5 | mpan2d 425 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝐴})) |
7 | elsni 3550 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴) | |
8 | 6, 7 | syl6 33 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → (𝑥 ∈ 𝐴 → 𝑥 = 𝐴)) |
9 | eleq1 2203 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐴 ∈ 𝐴)) | |
10 | 9 | biimpcd 158 | . . . 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 654 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → ¬ 𝑥 ∈ 𝐴) |
13 | 12 | eq0rdv 3412 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 ∅c0 3368 {csn 3532 Oncon0 4293 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-in 3082 df-ss 3089 df-nul 3369 df-sn 3538 df-uni 3745 df-tr 4035 df-iord 4296 df-on 4298 |
This theorem is referenced by: (None) |
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