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| Description: Lemma for ackbij2 10282. (Contributed by Stefan O'Rear, 18-Nov-2014.) | 
| Ref | Expression | 
|---|---|
| ackbij1lem1 | ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∩ suc 𝐴) = (𝐵 ∩ 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-suc 6390 | . . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
| 2 | 1 | ineq2i 4217 | . . 3 ⊢ (𝐵 ∩ suc 𝐴) = (𝐵 ∩ (𝐴 ∪ {𝐴})) | 
| 3 | indi 4284 | . . 3 ⊢ (𝐵 ∩ (𝐴 ∪ {𝐴})) = ((𝐵 ∩ 𝐴) ∪ (𝐵 ∩ {𝐴})) | |
| 4 | 2, 3 | eqtri 2765 | . 2 ⊢ (𝐵 ∩ suc 𝐴) = ((𝐵 ∩ 𝐴) ∪ (𝐵 ∩ {𝐴})) | 
| 5 | disjsn 4711 | . . . . 5 ⊢ ((𝐵 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐵) | |
| 6 | 5 | biimpri 228 | . . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∩ {𝐴}) = ∅) | 
| 7 | 6 | uneq2d 4168 | . . 3 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐵 ∩ 𝐴) ∪ (𝐵 ∩ {𝐴})) = ((𝐵 ∩ 𝐴) ∪ ∅)) | 
| 8 | un0 4394 | . . 3 ⊢ ((𝐵 ∩ 𝐴) ∪ ∅) = (𝐵 ∩ 𝐴) | |
| 9 | 7, 8 | eqtrdi 2793 | . 2 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐵 ∩ 𝐴) ∪ (𝐵 ∩ {𝐴})) = (𝐵 ∩ 𝐴)) | 
| 10 | 4, 9 | eqtrid 2789 | 1 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∩ suc 𝐴) = (𝐵 ∩ 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 ∩ cin 3950 ∅c0 4333 {csn 4626 suc csuc 6386 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-nul 4334 df-sn 4627 df-suc 6390 | 
| This theorem is referenced by: ackbij1lem15 10273 ackbij1lem16 10274 | 
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