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Mirrors > Home > ILE Home > Th. List > ss1o0el1 | GIF version |
Description: A subclass of {∅} contains the empty set if and only if it equals {∅}. (Contributed by BJ and Jim Kingdon, 9-Aug-2024.) |
Ref | Expression |
---|---|
ss1o0el1 | ⊢ (𝐴 ⊆ {∅} → (∅ ∈ 𝐴 ↔ 𝐴 = {∅})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex2 2728 | . . . 4 ⊢ (∅ ∈ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴) | |
2 | sssnm 3717 | . . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 ⊆ {∅} ↔ 𝐴 = {∅})) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (∅ ∈ 𝐴 → (𝐴 ⊆ {∅} ↔ 𝐴 = {∅})) |
4 | 3 | biimpcd 158 | . 2 ⊢ (𝐴 ⊆ {∅} → (∅ ∈ 𝐴 → 𝐴 = {∅})) |
5 | 0ex 4091 | . . . 4 ⊢ ∅ ∈ V | |
6 | 5 | snid 3591 | . . 3 ⊢ ∅ ∈ {∅} |
7 | eleq2 2221 | . . 3 ⊢ (𝐴 = {∅} → (∅ ∈ 𝐴 ↔ ∅ ∈ {∅})) | |
8 | 6, 7 | mpbiri 167 | . 2 ⊢ (𝐴 = {∅} → ∅ ∈ 𝐴) |
9 | 4, 8 | impbid1 141 | 1 ⊢ (𝐴 ⊆ {∅} → (∅ ∈ 𝐴 ↔ 𝐴 = {∅})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1335 ∃wex 1472 ∈ wcel 2128 ⊆ wss 3102 ∅c0 3394 {csn 3560 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 ax-nul 4090 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-dif 3104 df-in 3108 df-ss 3115 df-nul 3395 df-sn 3566 |
This theorem is referenced by: exmid01 4159 ss1o0el1o 6854 |
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