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| Mirrors > Home > MPE Home > Th. List > Mathboxes > mosssn2 | Structured version Visualization version GIF version | ||
| Description: Two ways of expressing "at most one" element in a class. (Contributed by Zhi Wang, 23-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| mosssn2 | ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝐴 ⊆ {𝑦}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 19.45v 1992 | . 2 ⊢ (∃𝑦(𝐴 = ∅ ∨ 𝐴 = {𝑦}) ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦})) | |
| 2 | sssn 4825 | . . 3 ⊢ (𝐴 ⊆ {𝑦} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝑦})) | |
| 3 | 2 | exbii 1847 | . 2 ⊢ (∃𝑦 𝐴 ⊆ {𝑦} ↔ ∃𝑦(𝐴 = ∅ ∨ 𝐴 = {𝑦})) | 
| 4 | mo0sn 48740 | . 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦})) | |
| 5 | 1, 3, 4 | 3bitr4ri 304 | 1 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝐴 ⊆ {𝑦}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∨ wo 847 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ∃*wmo 2537 ⊆ wss 3950 ∅c0 4332 {csn 4625 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-v 3481 df-sbc 3788 df-dif 3953 df-ss 3967 df-nul 4333 df-sn 4626 | 
| This theorem is referenced by: subthinc 49117 | 
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