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| Description: In the Separation Scheme zfauscl 5297, 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 if that requirement is not met. (Contributed by NM, 8-Feb-2006.) (Proof shortened by BJ, 18-Nov-2023.) | 
| 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 5306 | . . . . . . 7 ⊢ ∅ ∈ V | |
| 3 | 2 | snnz 4775 | . . . . . 6 ⊢ {∅} ≠ ∅ | 
| 4 | 1, 3 | eqnetri 3010 | . . . . 5 ⊢ 𝐴 ≠ ∅ | 
| 5 | n0 4352 | . . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
| 6 | 4, 5 | mpbi 230 | . . . 4 ⊢ ∃𝑥 𝑥 ∈ 𝐴 | 
| 7 | pm5.19 386 | . . . . 5 ⊢ ¬ (𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑦) | |
| 8 | ibar 528 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) | |
| 9 | notzfaus.2 | . . . . . . 7 ⊢ (𝜑 ↔ ¬ 𝑥 ∈ 𝑦) | |
| 10 | 8, 9 | bitr3di 286 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ¬ 𝑥 ∈ 𝑦)) | 
| 11 | 10 | bibi2d 342 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑦))) | 
| 12 | 7, 11 | mtbiri 327 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) | 
| 13 | 6, 12 | eximii 1836 | . . 3 ⊢ ∃𝑥 ¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) | 
| 14 | exnal 1826 | . . 3 ⊢ (∃𝑥 ¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) | |
| 15 | 13, 14 | mpbi 230 | . 2 ⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) | 
| 16 | 15 | nex 1799 | 1 ⊢ ¬ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∀wal 1537 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ≠ wne 2939 ∅c0 4332 {csn 4625 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-nul 5305 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-v 3481 df-dif 3953 df-nul 4333 df-sn 4626 | 
| This theorem is referenced by: (None) | 
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