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| Mirrors > Home > MPE Home > Th. List > eqsnuniex | Structured version Visualization version GIF version | ||
| Description: If a class is equal to the singleton of its union, then its union exists. (Contributed by BTernaryTau, 24-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| eqsnuniex | ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | unieq 4918 | . . . . 5 ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 = ∪ {∪ 𝐴}) | |
| 2 | unieq 4918 | . . . . . 6 ⊢ ({∪ 𝐴} = ∅ → ∪ {∪ 𝐴} = ∪ ∅) | |
| 3 | uni0 4935 | . . . . . 6 ⊢ ∪ ∅ = ∅ | |
| 4 | 2, 3 | eqtrdi 2793 | . . . . 5 ⊢ ({∪ 𝐴} = ∅ → ∪ {∪ 𝐴} = ∅) | 
| 5 | 1, 4 | sylan9eq 2797 | . . . 4 ⊢ ((𝐴 = {∪ 𝐴} ∧ {∪ 𝐴} = ∅) → ∪ 𝐴 = ∅) | 
| 6 | 5 | sneqd 4638 | . . 3 ⊢ ((𝐴 = {∪ 𝐴} ∧ {∪ 𝐴} = ∅) → {∪ 𝐴} = {∅}) | 
| 7 | 0inp0 5359 | . . . 4 ⊢ ({∪ 𝐴} = ∅ → ¬ {∪ 𝐴} = {∅}) | |
| 8 | 7 | adantl 481 | . . 3 ⊢ ((𝐴 = {∪ 𝐴} ∧ {∪ 𝐴} = ∅) → ¬ {∪ 𝐴} = {∅}) | 
| 9 | 6, 8 | pm2.65da 817 | . 2 ⊢ (𝐴 = {∪ 𝐴} → ¬ {∪ 𝐴} = ∅) | 
| 10 | snprc 4717 | . . . 4 ⊢ (¬ ∪ 𝐴 ∈ V ↔ {∪ 𝐴} = ∅) | |
| 11 | 10 | bicomi 224 | . . 3 ⊢ ({∪ 𝐴} = ∅ ↔ ¬ ∪ 𝐴 ∈ V) | 
| 12 | 11 | con2bii 357 | . 2 ⊢ (∪ 𝐴 ∈ V ↔ ¬ {∪ 𝐴} = ∅) | 
| 13 | 9, 12 | sylibr 234 | 1 ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 {csn 4626 ∪ cuni 4907 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-ext 2708 ax-nul 5306 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-v 3482 df-dif 3954 df-ss 3968 df-nul 4334 df-sn 4627 df-uni 4908 | 
| This theorem is referenced by: en1b 9065 en1uniel 9069 | 
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