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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 4816 | . . . . 5 ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 = ∪ {∪ 𝐴}) | |
2 | unieq 4816 | . . . . . 6 ⊢ ({∪ 𝐴} = ∅ → ∪ {∪ 𝐴} = ∪ ∅) | |
3 | uni0 4835 | . . . . . 6 ⊢ ∪ ∅ = ∅ | |
4 | 2, 3 | eqtrdi 2787 | . . . . 5 ⊢ ({∪ 𝐴} = ∅ → ∪ {∪ 𝐴} = ∅) |
5 | 1, 4 | sylan9eq 2791 | . . . 4 ⊢ ((𝐴 = {∪ 𝐴} ∧ {∪ 𝐴} = ∅) → ∪ 𝐴 = ∅) |
6 | 5 | sneqd 4539 | . . 3 ⊢ ((𝐴 = {∪ 𝐴} ∧ {∪ 𝐴} = ∅) → {∪ 𝐴} = {∅}) |
7 | 0inp0 5235 | . . . 4 ⊢ ({∪ 𝐴} = ∅ → ¬ {∪ 𝐴} = {∅}) | |
8 | 7 | adantl 485 | . . 3 ⊢ ((𝐴 = {∪ 𝐴} ∧ {∪ 𝐴} = ∅) → ¬ {∪ 𝐴} = {∅}) |
9 | 6, 8 | pm2.65da 817 | . 2 ⊢ (𝐴 = {∪ 𝐴} → ¬ {∪ 𝐴} = ∅) |
10 | snprc 4619 | . . . 4 ⊢ (¬ ∪ 𝐴 ∈ V ↔ {∪ 𝐴} = ∅) | |
11 | 10 | bicomi 227 | . . 3 ⊢ ({∪ 𝐴} = ∅ ↔ ¬ ∪ 𝐴 ∈ V) |
12 | 11 | con2bii 361 | . 2 ⊢ (∪ 𝐴 ∈ V ↔ ¬ {∪ 𝐴} = ∅) |
13 | 9, 12 | sylibr 237 | 1 ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 Vcvv 3398 ∅c0 4223 {csn 4527 ∪ cuni 4805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-11 2160 ax-ext 2708 ax-nul 5184 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-ral 3056 df-rex 3057 df-v 3400 df-dif 3856 df-in 3860 df-ss 3870 df-nul 4224 df-sn 4528 df-uni 4806 |
This theorem is referenced by: en1b 8678 en1uniel 8683 |
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