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Mirrors > Home > MPE Home > Th. List > notzfaus | Structured version Visualization version GIF version |
Description: In the Separation Scheme zfauscl 4711, we require that 𝑦 not occur in 𝜑 (which can be generalized to "not be free in"). Here we show special cases of 𝐴 and 𝜑 that result in a contradiction by violating this requirement. (Contributed by NM, 8-Feb-2006.) |
Ref | Expression |
---|---|
notzfaus.1 | ⊢ 𝐴 = {∅} |
notzfaus.2 | ⊢ (𝜑 ↔ ¬ 𝑥 ∈ 𝑦) |
Ref | Expression |
---|---|
notzfaus | ⊢ ¬ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notzfaus.1 | . . . . . 6 ⊢ 𝐴 = {∅} | |
2 | 0ex 4718 | . . . . . . 7 ⊢ ∅ ∈ V | |
3 | 2 | snnz 4252 | . . . . . 6 ⊢ {∅} ≠ ∅ |
4 | 1, 3 | eqnetri 2852 | . . . . 5 ⊢ 𝐴 ≠ ∅ |
5 | n0 3890 | . . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
6 | 4, 5 | mpbi 219 | . . . 4 ⊢ ∃𝑥 𝑥 ∈ 𝐴 |
7 | biimt 349 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑦))) | |
8 | iman 439 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑦) ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝑦)) | |
9 | notzfaus.2 | . . . . . . . 8 ⊢ (𝜑 ↔ ¬ 𝑥 ∈ 𝑦) | |
10 | 9 | anbi2i 726 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝑦)) |
11 | 8, 10 | xchbinxr 324 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑦) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
12 | 7, 11 | syl6bb 275 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝑦 ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))) |
13 | xor3 371 | . . . . 5 ⊢ (¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑦 ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))) | |
14 | 12, 13 | sylibr 223 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) |
15 | 6, 14 | eximii 1754 | . . 3 ⊢ ∃𝑥 ¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
16 | exnal 1744 | . . 3 ⊢ (∃𝑥 ¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) | |
17 | 15, 16 | mpbi 219 | . 2 ⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
18 | 17 | nex 1722 | 1 ⊢ ¬ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 {csn 4125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-dif 3543 df-nul 3875 df-sn 4126 |
This theorem is referenced by: (None) |
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