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Mirrors > Home > MPE Home > Th. List > axnulALT | Structured version Visualization version GIF version |
Description: Alternate proof of axnul 5200, proved from propositional calculus, ax-gen 1787, ax-4 1801, sp 2172, and ax-rep 5181. To check this, replace sp 2172 with the obsolete axiom ax-c5 35899 in the proof of axnulALT 5199 and type the Metamath program "MM> SHOW TRACE_BACK 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 5181 | . . 3 ⊢ (∀𝑤∃𝑥∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) → ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥))) | |
2 | sp 2172 | . . . . . 6 ⊢ (∀𝑥 ¬ ∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) → ¬ ∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥)) | |
3 | 2 | con2i 141 | . . . . 5 ⊢ (∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) → ¬ ∀𝑥 ¬ ∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥)) |
4 | df-ex 1772 | . . . . 5 ⊢ (∃𝑥∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) ↔ ¬ ∀𝑥 ¬ ∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥)) | |
5 | 3, 4 | sylibr 235 | . . . 4 ⊢ (∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) → ∃𝑥∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥)) |
6 | fal 1542 | . . . . . 6 ⊢ ¬ ⊥ | |
7 | sp 2172 | . . . . . 6 ⊢ (∀𝑥⊥ → ⊥) | |
8 | 6, 7 | mto 198 | . . . . 5 ⊢ ¬ ∀𝑥⊥ |
9 | 8 | pm2.21i 119 | . . . 4 ⊢ (∀𝑥⊥ → 𝑦 = 𝑥) |
10 | 5, 9 | mpg 1789 | . . 3 ⊢ ∃𝑥∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) |
11 | 1, 10 | mpg 1789 | . 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)) |
12 | 8 | intnan 487 | . . . . . 6 ⊢ ¬ (𝑤 ∈ 𝑧 ∧ ∀𝑥⊥) |
13 | 12 | nex 1792 | . . . . 5 ⊢ ¬ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥) |
14 | 13 | nbn 374 | . . . 4 ⊢ (¬ 𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥))) |
15 | 14 | albii 1811 | . . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥))) |
16 | 15 | exbii 1839 | . 2 ⊢ (∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥))) |
17 | 11, 16 | mpbir 232 | 1 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1526 = wceq 1528 ⊥wfal 1540 ∃wex 1771 ∈ wcel 2105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-12 2167 ax-rep 5181 |
This theorem depends on definitions: df-bi 208 df-an 397 df-tru 1531 df-fal 1541 df-ex 1772 |
This theorem is referenced by: (None) |
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