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Mirrors > Home > MPE Home > Th. List > 2pwuninel | Structured version Visualization version GIF version |
Description: The power set of the power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by NM, 27-Jun-2008.) |
Ref | Expression |
---|---|
2pwuninel | ⊢ ¬ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sdomirr 9153 | . . 3 ⊢ ¬ 𝒫 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴 | |
2 | elssuni 4942 | . . . 4 ⊢ (𝒫 𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ⊆ ∪ 𝐴) | |
3 | ssdomg 9039 | . . . . 5 ⊢ (∪ 𝐴 ∈ V → (𝒫 𝒫 ∪ 𝐴 ⊆ ∪ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ≼ ∪ 𝐴)) | |
4 | canth2g 9170 | . . . . . 6 ⊢ (∪ 𝐴 ∈ V → ∪ 𝐴 ≺ 𝒫 ∪ 𝐴) | |
5 | pwexb 7785 | . . . . . . 7 ⊢ (∪ 𝐴 ∈ V ↔ 𝒫 ∪ 𝐴 ∈ V) | |
6 | canth2g 9170 | . . . . . . 7 ⊢ (𝒫 ∪ 𝐴 ∈ V → 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴) | |
7 | 5, 6 | sylbi 217 | . . . . . 6 ⊢ (∪ 𝐴 ∈ V → 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴) |
8 | sdomtr 9154 | . . . . . 6 ⊢ ((∪ 𝐴 ≺ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴) → ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴) | |
9 | 4, 7, 8 | syl2anc 584 | . . . . 5 ⊢ (∪ 𝐴 ∈ V → ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴) |
10 | domsdomtr 9151 | . . . . . 6 ⊢ ((𝒫 𝒫 ∪ 𝐴 ≼ ∪ 𝐴 ∧ ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴) → 𝒫 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴) | |
11 | 10 | ex 412 | . . . . 5 ⊢ (𝒫 𝒫 ∪ 𝐴 ≼ ∪ 𝐴 → (∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴)) |
12 | 3, 9, 11 | syl6ci 71 | . . . 4 ⊢ (∪ 𝐴 ∈ V → (𝒫 𝒫 ∪ 𝐴 ⊆ ∪ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴)) |
13 | 2, 12 | syl5 34 | . . 3 ⊢ (∪ 𝐴 ∈ V → (𝒫 𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ≺ 𝒫 𝒫 ∪ 𝐴)) |
14 | 1, 13 | mtoi 199 | . 2 ⊢ (∪ 𝐴 ∈ V → ¬ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴) |
15 | elex 3499 | . . . 4 ⊢ (𝒫 𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 𝒫 ∪ 𝐴 ∈ V) | |
16 | pwexb 7785 | . . . . 5 ⊢ (𝒫 ∪ 𝐴 ∈ V ↔ 𝒫 𝒫 ∪ 𝐴 ∈ V) | |
17 | 5, 16 | bitri 275 | . . . 4 ⊢ (∪ 𝐴 ∈ V ↔ 𝒫 𝒫 ∪ 𝐴 ∈ V) |
18 | 15, 17 | sylibr 234 | . . 3 ⊢ (𝒫 𝒫 ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 ∈ V) |
19 | 18 | con3i 154 | . 2 ⊢ (¬ ∪ 𝐴 ∈ V → ¬ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴) |
20 | 14, 19 | pm2.61i 182 | 1 ⊢ ¬ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 𝒫 cpw 4605 ∪ cuni 4912 class class class wbr 5148 ≼ cdom 8982 ≺ csdm 8983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 |
This theorem is referenced by: mnfnre 11302 |
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