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Mirrors > Home > MPE Home > Th. List > ackbij1lem1 | Structured version Visualization version GIF version |
Description: Lemma for ackbij2 10187. (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 6327 | . . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
2 | 1 | ineq2i 4173 | . . 3 ⊢ (𝐵 ∩ suc 𝐴) = (𝐵 ∩ (𝐴 ∪ {𝐴})) |
3 | indi 4237 | . . 3 ⊢ (𝐵 ∩ (𝐴 ∪ {𝐴})) = ((𝐵 ∩ 𝐴) ∪ (𝐵 ∩ {𝐴})) | |
4 | 2, 3 | eqtri 2761 | . 2 ⊢ (𝐵 ∩ suc 𝐴) = ((𝐵 ∩ 𝐴) ∪ (𝐵 ∩ {𝐴})) |
5 | disjsn 4676 | . . . . 5 ⊢ ((𝐵 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐵) | |
6 | 5 | biimpri 227 | . . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∩ {𝐴}) = ∅) |
7 | 6 | uneq2d 4127 | . . 3 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐵 ∩ 𝐴) ∪ (𝐵 ∩ {𝐴})) = ((𝐵 ∩ 𝐴) ∪ ∅)) |
8 | un0 4354 | . . 3 ⊢ ((𝐵 ∩ 𝐴) ∪ ∅) = (𝐵 ∩ 𝐴) | |
9 | 7, 8 | eqtrdi 2789 | . 2 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐵 ∩ 𝐴) ∪ (𝐵 ∩ {𝐴})) = (𝐵 ∩ 𝐴)) |
10 | 4, 9 | eqtrid 2785 | 1 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∩ suc 𝐴) = (𝐵 ∩ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∈ wcel 2107 ∪ cun 3912 ∩ cin 3913 ∅c0 4286 {csn 4590 suc csuc 6323 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-nul 4287 df-sn 4591 df-suc 6327 |
This theorem is referenced by: ackbij1lem15 10178 ackbij1lem16 10179 |
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