Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ackbij1lem1 | Structured version Visualization version GIF version |
Description: Lemma for ackbij2 8948. (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 5646 | . . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
2 | 1 | ineq2i 3773 | . . 3 ⊢ (𝐵 ∩ suc 𝐴) = (𝐵 ∩ (𝐴 ∪ {𝐴})) |
3 | indi 3832 | . . 3 ⊢ (𝐵 ∩ (𝐴 ∪ {𝐴})) = ((𝐵 ∩ 𝐴) ∪ (𝐵 ∩ {𝐴})) | |
4 | 2, 3 | eqtri 2632 | . 2 ⊢ (𝐵 ∩ suc 𝐴) = ((𝐵 ∩ 𝐴) ∪ (𝐵 ∩ {𝐴})) |
5 | disjsn 4192 | . . . . 5 ⊢ ((𝐵 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐵) | |
6 | 5 | biimpri 217 | . . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∩ {𝐴}) = ∅) |
7 | 6 | uneq2d 3729 | . . 3 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐵 ∩ 𝐴) ∪ (𝐵 ∩ {𝐴})) = ((𝐵 ∩ 𝐴) ∪ ∅)) |
8 | un0 3919 | . . 3 ⊢ ((𝐵 ∩ 𝐴) ∪ ∅) = (𝐵 ∩ 𝐴) | |
9 | 7, 8 | syl6eq 2660 | . 2 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐵 ∩ 𝐴) ∪ (𝐵 ∩ {𝐴})) = (𝐵 ∩ 𝐴)) |
10 | 4, 9 | syl5eq 2656 | 1 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∩ suc 𝐴) = (𝐵 ∩ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 ∩ cin 3539 ∅c0 3874 {csn 4125 suc csuc 5642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-nul 3875 df-sn 4126 df-suc 5646 |
This theorem is referenced by: ackbij1lem15 8939 ackbij1lem16 8940 |
Copyright terms: Public domain | W3C validator |