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Mirrors > Home > MPE Home > Th. List > ackbij1lem1 | Structured version Visualization version GIF version |
Description: Lemma for ackbij2 9654. (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 6165 | . . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
2 | 1 | ineq2i 4136 | . . 3 ⊢ (𝐵 ∩ suc 𝐴) = (𝐵 ∩ (𝐴 ∪ {𝐴})) |
3 | indi 4200 | . . 3 ⊢ (𝐵 ∩ (𝐴 ∪ {𝐴})) = ((𝐵 ∩ 𝐴) ∪ (𝐵 ∩ {𝐴})) | |
4 | 2, 3 | eqtri 2821 | . 2 ⊢ (𝐵 ∩ suc 𝐴) = ((𝐵 ∩ 𝐴) ∪ (𝐵 ∩ {𝐴})) |
5 | disjsn 4607 | . . . . 5 ⊢ ((𝐵 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐵) | |
6 | 5 | biimpri 231 | . . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∩ {𝐴}) = ∅) |
7 | 6 | uneq2d 4090 | . . 3 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐵 ∩ 𝐴) ∪ (𝐵 ∩ {𝐴})) = ((𝐵 ∩ 𝐴) ∪ ∅)) |
8 | un0 4298 | . . 3 ⊢ ((𝐵 ∩ 𝐴) ∪ ∅) = (𝐵 ∩ 𝐴) | |
9 | 7, 8 | eqtrdi 2849 | . 2 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐵 ∩ 𝐴) ∪ (𝐵 ∩ {𝐴})) = (𝐵 ∩ 𝐴)) |
10 | 4, 9 | syl5eq 2845 | 1 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∩ suc 𝐴) = (𝐵 ∩ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1538 ∈ wcel 2111 ∪ cun 3879 ∩ cin 3880 ∅c0 4243 {csn 4525 suc csuc 6161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-nul 4244 df-sn 4526 df-suc 6165 |
This theorem is referenced by: ackbij1lem15 9645 ackbij1lem16 9646 |
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