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Mirrors > Home > MPE Home > Th. List > pwuninel2 | Structured version Visualization version GIF version |
Description: Direct proof of pwuninel 8210 avoiding functions and thus several ZF axioms. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Ref | Expression |
---|---|
pwuninel2 | ⊢ (∪ 𝐴 ∈ 𝑉 → ¬ 𝒫 ∪ 𝐴 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwnss 5310 | . 2 ⊢ (∪ 𝐴 ∈ 𝑉 → ¬ 𝒫 ∪ 𝐴 ⊆ ∪ 𝐴) | |
2 | elssuni 4902 | . 2 ⊢ (𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 ∪ 𝐴 ⊆ ∪ 𝐴) | |
3 | 1, 2 | nsyl 140 | 1 ⊢ (∪ 𝐴 ∈ 𝑉 → ¬ 𝒫 ∪ 𝐴 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2107 ⊆ wss 3914 𝒫 cpw 4564 ∪ cuni 4869 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5260 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3449 df-in 3921 df-ss 3931 df-pw 4566 df-uni 4870 |
This theorem is referenced by: pwuninel 8210 |
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