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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 4914 | . . . . 5 ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 = ∪ {∪ 𝐴}) | |
2 | unieq 4914 | . . . . . 6 ⊢ ({∪ 𝐴} = ∅ → ∪ {∪ 𝐴} = ∪ ∅) | |
3 | uni0 4933 | . . . . . 6 ⊢ ∪ ∅ = ∅ | |
4 | 2, 3 | eqtrdi 2781 | . . . . 5 ⊢ ({∪ 𝐴} = ∅ → ∪ {∪ 𝐴} = ∅) |
5 | 1, 4 | sylan9eq 2785 | . . . 4 ⊢ ((𝐴 = {∪ 𝐴} ∧ {∪ 𝐴} = ∅) → ∪ 𝐴 = ∅) |
6 | 5 | sneqd 4636 | . . 3 ⊢ ((𝐴 = {∪ 𝐴} ∧ {∪ 𝐴} = ∅) → {∪ 𝐴} = {∅}) |
7 | 0inp0 5353 | . . . 4 ⊢ ({∪ 𝐴} = ∅ → ¬ {∪ 𝐴} = {∅}) | |
8 | 7 | adantl 480 | . . 3 ⊢ ((𝐴 = {∪ 𝐴} ∧ {∪ 𝐴} = ∅) → ¬ {∪ 𝐴} = {∅}) |
9 | 6, 8 | pm2.65da 815 | . 2 ⊢ (𝐴 = {∪ 𝐴} → ¬ {∪ 𝐴} = ∅) |
10 | snprc 4717 | . . . 4 ⊢ (¬ ∪ 𝐴 ∈ V ↔ {∪ 𝐴} = ∅) | |
11 | 10 | bicomi 223 | . . 3 ⊢ ({∪ 𝐴} = ∅ ↔ ¬ ∪ 𝐴 ∈ V) |
12 | 11 | con2bii 356 | . 2 ⊢ (∪ 𝐴 ∈ V ↔ ¬ {∪ 𝐴} = ∅) |
13 | 9, 12 | sylibr 233 | 1 ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ∅c0 4318 {csn 4624 ∪ cuni 4903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-11 2146 ax-ext 2696 ax-nul 5301 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2931 df-ral 3052 df-rex 3061 df-v 3465 df-dif 3942 df-ss 3956 df-nul 4319 df-sn 4625 df-uni 4904 |
This theorem is referenced by: en1b 9046 en1uniel 9051 |
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