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Mirrors > Home > MPE Home > Th. List > ackbij1lem2 | Structured version Visualization version GIF version |
Description: Lemma for ackbij2 10280. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
Ref | Expression |
---|---|
ackbij1lem2 | ⊢ (𝐴 ∈ 𝐵 → (𝐵 ∩ suc 𝐴) = ({𝐴} ∪ (𝐵 ∩ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-suc 6392 | . . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
2 | 1 | ineq2i 4225 | . . 3 ⊢ (𝐵 ∩ suc 𝐴) = (𝐵 ∩ (𝐴 ∪ {𝐴})) |
3 | indi 4290 | . . 3 ⊢ (𝐵 ∩ (𝐴 ∪ {𝐴})) = ((𝐵 ∩ 𝐴) ∪ (𝐵 ∩ {𝐴})) | |
4 | uncom 4168 | . . 3 ⊢ ((𝐵 ∩ 𝐴) ∪ (𝐵 ∩ {𝐴})) = ((𝐵 ∩ {𝐴}) ∪ (𝐵 ∩ 𝐴)) | |
5 | 2, 3, 4 | 3eqtri 2767 | . 2 ⊢ (𝐵 ∩ suc 𝐴) = ((𝐵 ∩ {𝐴}) ∪ (𝐵 ∩ 𝐴)) |
6 | snssi 4813 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) | |
7 | sseqin2 4231 | . . . 4 ⊢ ({𝐴} ⊆ 𝐵 ↔ (𝐵 ∩ {𝐴}) = {𝐴}) | |
8 | 6, 7 | sylib 218 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐵 ∩ {𝐴}) = {𝐴}) |
9 | 8 | uneq1d 4177 | . 2 ⊢ (𝐴 ∈ 𝐵 → ((𝐵 ∩ {𝐴}) ∪ (𝐵 ∩ 𝐴)) = ({𝐴} ∪ (𝐵 ∩ 𝐴))) |
10 | 5, 9 | eqtrid 2787 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝐵 ∩ suc 𝐴) = ({𝐴} ∪ (𝐵 ∩ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 ∩ cin 3962 ⊆ wss 3963 {csn 4631 suc csuc 6388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-un 3968 df-in 3970 df-ss 3980 df-sn 4632 df-suc 6392 |
This theorem is referenced by: ackbij1lem15 10271 ackbij1lem16 10272 |
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