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Mirrors > Home > MPE Home > Th. List > notzfaus | Structured version Visualization version GIF version |
Description: In the Separation Scheme zfauscl 5179, 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 5185 | . . . . . . 7 ⊢ ∅ ∈ V | |
3 | 2 | snnz 4678 | . . . . . 6 ⊢ {∅} ≠ ∅ |
4 | 1, 3 | eqnetri 3002 | . . . . 5 ⊢ 𝐴 ≠ ∅ |
5 | n0 4247 | . . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
6 | 4, 5 | mpbi 233 | . . . 4 ⊢ ∃𝑥 𝑥 ∈ 𝐴 |
7 | pm5.19 391 | . . . . 5 ⊢ ¬ (𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑦) | |
8 | ibar 532 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) | |
9 | notzfaus.2 | . . . . . . 7 ⊢ (𝜑 ↔ ¬ 𝑥 ∈ 𝑦) | |
10 | 8, 9 | bitr3di 289 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ¬ 𝑥 ∈ 𝑦)) |
11 | 10 | bibi2d 346 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑦))) |
12 | 7, 11 | mtbiri 330 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) |
13 | 6, 12 | eximii 1844 | . . 3 ⊢ ∃𝑥 ¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
14 | exnal 1834 | . . 3 ⊢ (∃𝑥 ¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) | |
15 | 13, 14 | mpbi 233 | . 2 ⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
16 | 15 | nex 1808 | 1 ⊢ ¬ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 399 ∀wal 1541 = wceq 1543 ∃wex 1787 ∈ wcel 2112 ≠ wne 2932 ∅c0 4223 {csn 4527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-nul 5184 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-v 3400 df-dif 3856 df-nul 4224 df-sn 4528 |
This theorem is referenced by: (None) |
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