Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > inpr0 | Structured version Visualization version GIF version |
Description: Rewrite an empty intersection with a pair. (Contributed by Thierry Arnoux, 20-Nov-2023.) |
Ref | Expression |
---|---|
inpr0 | ⊢ ((𝐴 ∩ {𝐵, 𝐶}) = ∅ ↔ (¬ 𝐵 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.26 3169 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶) ↔ (∀𝑥 ∈ 𝐴 𝑥 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≠ 𝐶)) | |
2 | nelpr 30291 | . . . . . 6 ⊢ (𝑥 ∈ V → (¬ 𝑥 ∈ {𝐵, 𝐶} ↔ (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶))) | |
3 | 2 | elv 3496 | . . . . 5 ⊢ (¬ 𝑥 ∈ {𝐵, 𝐶} ↔ (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶)) |
4 | 3 | imbi2i 338 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ {𝐵, 𝐶}) ↔ (𝑥 ∈ 𝐴 → (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶))) |
5 | 4 | albii 1819 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ {𝐵, 𝐶}) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶))) |
6 | disj1 4394 | . . 3 ⊢ ((𝐴 ∩ {𝐵, 𝐶}) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ {𝐵, 𝐶})) | |
7 | df-ral 3142 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶))) | |
8 | 5, 6, 7 | 3bitr4i 305 | . 2 ⊢ ((𝐴 ∩ {𝐵, 𝐶}) = ∅ ↔ ∀𝑥 ∈ 𝐴 (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶)) |
9 | nelb 3267 | . . 3 ⊢ (¬ 𝐵 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ≠ 𝐵) | |
10 | nelb 3267 | . . 3 ⊢ (¬ 𝐶 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ≠ 𝐶) | |
11 | 9, 10 | anbi12i 628 | . 2 ⊢ ((¬ 𝐵 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) ↔ (∀𝑥 ∈ 𝐴 𝑥 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≠ 𝐶)) |
12 | 1, 8, 11 | 3bitr4i 305 | 1 ⊢ ((𝐴 ∩ {𝐵, 𝐶}) = ∅ ↔ (¬ 𝐵 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1534 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 ∀wral 3137 Vcvv 3491 ∩ cin 3928 ∅c0 4284 {cpr 4562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-nul 4285 df-sn 4561 df-pr 4563 |
This theorem is referenced by: (None) |
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