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Mirrors > Home > MPE Home > Th. List > disjOLD | Structured version Visualization version GIF version |
Description: Obsolete version of disj 4337 as of 28-Jun-2024. (Contributed by NM, 17-Feb-2004.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
disjOLD | ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-in 3861 | . . . 4 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} | |
2 | 1 | eqeq1i 2764 | . . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = ∅) |
3 | abeq1 2884 | . . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = ∅ ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅)) | |
4 | imnan 404 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
5 | noel 4226 | . . . . . 6 ⊢ ¬ 𝑥 ∈ ∅ | |
6 | 5 | nbn 377 | . . . . 5 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅)) |
7 | 4, 6 | bitr2i 279 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅) ↔ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) |
8 | 7 | albii 1822 | . . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ ∅) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) |
9 | 2, 3, 8 | 3bitri 301 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) |
10 | df-ral 3073 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) | |
11 | 9, 10 | bitr4i 281 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∀wal 1537 = wceq 1539 ∈ wcel 2112 {cab 2736 ∀wral 3068 ∩ cin 3853 ∅c0 4221 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-ral 3073 df-dif 3857 df-in 3861 df-nul 4222 |
This theorem is referenced by: (None) |
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