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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > snsssng | Structured version Visualization version GIF version |
Description: If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.) (Revised by Thierry Arnoux, 11-Apr-2024.) |
Ref | Expression |
---|---|
snsssng | ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ⊆ {𝐵}) → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sssn 4784 | . 2 ⊢ ({𝐴} ⊆ {𝐵} ↔ ({𝐴} = ∅ ∨ {𝐴} = {𝐵})) | |
2 | snnzg 4733 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅) | |
3 | 2 | neneqd 2946 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ¬ {𝐴} = ∅) |
4 | 3 | pm2.21d 121 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = ∅ → 𝐴 = 𝐵)) |
5 | sneqrg 4795 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵)) | |
6 | 4, 5 | jaod 857 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (({𝐴} = ∅ ∨ {𝐴} = {𝐵}) → 𝐴 = 𝐵)) |
7 | 6 | imp 407 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ({𝐴} = ∅ ∨ {𝐴} = {𝐵})) → 𝐴 = 𝐵) |
8 | 1, 7 | sylan2b 594 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ⊆ {𝐵}) → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ⊆ wss 3908 ∅c0 4280 {csn 4584 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-v 3445 df-dif 3911 df-in 3915 df-ss 3925 df-nul 4281 df-sn 4585 |
This theorem is referenced by: (None) |
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