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Mirrors > Home > MPE Home > Th. List > axnulALT | Structured version Visualization version GIF version |
Description: Alternate proof of axnul 5014, proved from propositional calculus, ax-gen 1894, ax-4 1908, sp 2224, and ax-rep 4996. To check this, replace sp 2224 with the obsolete axiom ax-c5 34953 in the proof of axnulALT 5013 and type the Metamath program "MM> SHOW TRACEBACK axnulALT / AXIOMS" command. (Contributed by Jeff Hoffman, 3-Feb-2008.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axnulALT | ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-rep 4996 | . . 3 ⊢ (∀𝑤∃𝑥∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) → ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥))) | |
2 | sp 2224 | . . . . . 6 ⊢ (∀𝑥 ¬ ∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) → ¬ ∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥)) | |
3 | 2 | con2i 137 | . . . . 5 ⊢ (∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) → ¬ ∀𝑥 ¬ ∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥)) |
4 | df-ex 1879 | . . . . 5 ⊢ (∃𝑥∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) ↔ ¬ ∀𝑥 ¬ ∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥)) | |
5 | 3, 4 | sylibr 226 | . . . 4 ⊢ (∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) → ∃𝑥∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥)) |
6 | fal 1671 | . . . . . 6 ⊢ ¬ ⊥ | |
7 | sp 2224 | . . . . . 6 ⊢ (∀𝑥⊥ → ⊥) | |
8 | 6, 7 | mto 189 | . . . . 5 ⊢ ¬ ∀𝑥⊥ |
9 | 8 | pm2.21i 117 | . . . 4 ⊢ (∀𝑥⊥ → 𝑦 = 𝑥) |
10 | 5, 9 | mpg 1896 | . . 3 ⊢ ∃𝑥∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) |
11 | 1, 10 | mpg 1896 | . 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)) |
12 | 8 | intnan 482 | . . . . . 6 ⊢ ¬ (𝑤 ∈ 𝑧 ∧ ∀𝑥⊥) |
13 | 12 | nex 1899 | . . . . 5 ⊢ ¬ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥) |
14 | 13 | nbn 364 | . . . 4 ⊢ (¬ 𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥))) |
15 | 14 | albii 1918 | . . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥))) |
16 | 15 | exbii 1947 | . 2 ⊢ (∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥))) |
17 | 11, 16 | mpbir 223 | 1 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∀wal 1654 = wceq 1656 ⊥wfal 1669 ∃wex 1878 ∈ wcel 2164 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-12 2220 ax-rep 4996 |
This theorem depends on definitions: df-bi 199 df-an 387 df-tru 1660 df-fal 1670 df-ex 1879 |
This theorem is referenced by: (None) |
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