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Mirrors > Home > MPE Home > Th. List > axnulALT | Structured version Visualization version GIF version |
Description: Alternate proof of axnul 5298, proved from propositional calculus, ax-gen 1789, ax-4 1803, sp 2168, and ax-rep 5278. To check this, replace sp 2168 with the obsolete axiom ax-c5 38266 in the proof of axnulALT 5297 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 5278 | . . 3 ⊢ (∀𝑤∃𝑥∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) → ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥))) | |
2 | sp 2168 | . . . . . 6 ⊢ (∀𝑥 ¬ ∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) → ¬ ∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥)) | |
3 | 2 | con2i 139 | . . . . 5 ⊢ (∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) → ¬ ∀𝑥 ¬ ∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥)) |
4 | df-ex 1774 | . . . . 5 ⊢ (∃𝑥∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) ↔ ¬ ∀𝑥 ¬ ∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥)) | |
5 | 3, 4 | sylibr 233 | . . . 4 ⊢ (∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) → ∃𝑥∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥)) |
6 | fal 1547 | . . . . . 6 ⊢ ¬ ⊥ | |
7 | sp 2168 | . . . . . 6 ⊢ (∀𝑥⊥ → ⊥) | |
8 | 6, 7 | mto 196 | . . . . 5 ⊢ ¬ ∀𝑥⊥ |
9 | 8 | pm2.21i 119 | . . . 4 ⊢ (∀𝑥⊥ → 𝑦 = 𝑥) |
10 | 5, 9 | mpg 1791 | . . 3 ⊢ ∃𝑥∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) |
11 | 1, 10 | mpg 1791 | . 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)) |
12 | 8 | intnan 486 | . . . . . 6 ⊢ ¬ (𝑤 ∈ 𝑧 ∧ ∀𝑥⊥) |
13 | 12 | nex 1794 | . . . . 5 ⊢ ¬ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥) |
14 | 13 | nbn 372 | . . . 4 ⊢ (¬ 𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥))) |
15 | 14 | albii 1813 | . . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥))) |
16 | 15 | exbii 1842 | . 2 ⊢ (∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥))) |
17 | 11, 16 | mpbir 230 | 1 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1531 = wceq 1533 ⊥wfal 1545 ∃wex 1773 ∈ wcel 2098 |
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-12 2163 ax-rep 5278 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-fal 1546 df-ex 1774 |
This theorem is referenced by: (None) |
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