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Mirrors > Home > MPE Home > Th. List > notzfaus | Structured version Visualization version GIF version |
Description: In the Separation Scheme zfauscl 5196, 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 5202 | . . . . . . 7 ⊢ ∅ ∈ V | |
3 | 2 | snnz 4703 | . . . . . 6 ⊢ {∅} ≠ ∅ |
4 | 1, 3 | eqnetri 3083 | . . . . 5 ⊢ 𝐴 ≠ ∅ |
5 | n0 4307 | . . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
6 | 4, 5 | mpbi 231 | . . . 4 ⊢ ∃𝑥 𝑥 ∈ 𝐴 |
7 | pm5.19 388 | . . . . 5 ⊢ ¬ (𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑦) | |
8 | notzfaus.2 | . . . . . . 7 ⊢ (𝜑 ↔ ¬ 𝑥 ∈ 𝑦) | |
9 | ibar 529 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) | |
10 | 8, 9 | syl5rbbr 287 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ¬ 𝑥 ∈ 𝑦)) |
11 | 10 | bibi2d 344 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑦))) |
12 | 7, 11 | mtbiri 328 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) |
13 | 6, 12 | eximii 1828 | . . 3 ⊢ ∃𝑥 ¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
14 | exnal 1818 | . . 3 ⊢ (∃𝑥 ¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) | |
15 | 13, 14 | mpbi 231 | . 2 ⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
16 | 15 | nex 1792 | 1 ⊢ ¬ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 ∧ wa 396 ∀wal 1526 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ≠ wne 3013 ∅c0 4288 {csn 4557 |
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-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-nul 5201 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-v 3494 df-dif 3936 df-nul 4289 df-sn 4558 |
This theorem is referenced by: (None) |
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