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Mirrors > Home > MPE Home > Th. List > ackbij1lem2 | Structured version Visualization version GIF version |
Description: Lemma for ackbij2 10240. (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 6364 | . . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
2 | 1 | ineq2i 4204 | . . 3 ⊢ (𝐵 ∩ suc 𝐴) = (𝐵 ∩ (𝐴 ∪ {𝐴})) |
3 | indi 4268 | . . 3 ⊢ (𝐵 ∩ (𝐴 ∪ {𝐴})) = ((𝐵 ∩ 𝐴) ∪ (𝐵 ∩ {𝐴})) | |
4 | uncom 4148 | . . 3 ⊢ ((𝐵 ∩ 𝐴) ∪ (𝐵 ∩ {𝐴})) = ((𝐵 ∩ {𝐴}) ∪ (𝐵 ∩ 𝐴)) | |
5 | 2, 3, 4 | 3eqtri 2758 | . 2 ⊢ (𝐵 ∩ suc 𝐴) = ((𝐵 ∩ {𝐴}) ∪ (𝐵 ∩ 𝐴)) |
6 | snssi 4806 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) | |
7 | sseqin2 4210 | . . . 4 ⊢ ({𝐴} ⊆ 𝐵 ↔ (𝐵 ∩ {𝐴}) = {𝐴}) | |
8 | 6, 7 | sylib 217 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐵 ∩ {𝐴}) = {𝐴}) |
9 | 8 | uneq1d 4157 | . 2 ⊢ (𝐴 ∈ 𝐵 → ((𝐵 ∩ {𝐴}) ∪ (𝐵 ∩ 𝐴)) = ({𝐴} ∪ (𝐵 ∩ 𝐴))) |
10 | 5, 9 | eqtrid 2778 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝐵 ∩ suc 𝐴) = ({𝐴} ∪ (𝐵 ∩ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∪ cun 3941 ∩ cin 3942 ⊆ wss 3943 {csn 4623 suc csuc 6360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-rab 3427 df-v 3470 df-un 3948 df-in 3950 df-ss 3960 df-sn 4624 df-suc 6364 |
This theorem is referenced by: ackbij1lem15 10231 ackbij1lem16 10232 |
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