| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | sadval.a | . . 3
⊢ (𝜑 → 𝐴 ⊆
ℕ0) | 
| 2 |  | nn0ex 12534 | . . . 4
⊢
ℕ0 ∈ V | 
| 3 | 2 | elpw2 5333 | . . 3
⊢ (𝐴 ∈ 𝒫
ℕ0 ↔ 𝐴 ⊆
ℕ0) | 
| 4 | 1, 3 | sylibr 234 | . 2
⊢ (𝜑 → 𝐴 ∈ 𝒫
ℕ0) | 
| 5 |  | sadval.b | . . 3
⊢ (𝜑 → 𝐵 ⊆
ℕ0) | 
| 6 | 2 | elpw2 5333 | . . 3
⊢ (𝐵 ∈ 𝒫
ℕ0 ↔ 𝐵 ⊆
ℕ0) | 
| 7 | 5, 6 | sylibr 234 | . 2
⊢ (𝜑 → 𝐵 ∈ 𝒫
ℕ0) | 
| 8 |  | simpl 482 | . . . . . 6
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴) | 
| 9 | 8 | eleq2d 2826 | . . . . 5
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑘 ∈ 𝑥 ↔ 𝑘 ∈ 𝐴)) | 
| 10 |  | simpr 484 | . . . . . 6
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵) | 
| 11 | 10 | eleq2d 2826 | . . . . 5
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑘 ∈ 𝑦 ↔ 𝑘 ∈ 𝐵)) | 
| 12 |  | simp1l 1197 | . . . . . . . . . . . . 13
⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑐 ∈ 2o ∧ 𝑚 ∈ ℕ0)
→ 𝑥 = 𝐴) | 
| 13 | 12 | eleq2d 2826 | . . . . . . . . . . . 12
⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑐 ∈ 2o ∧ 𝑚 ∈ ℕ0)
→ (𝑚 ∈ 𝑥 ↔ 𝑚 ∈ 𝐴)) | 
| 14 |  | simp1r 1198 | . . . . . . . . . . . . 13
⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑐 ∈ 2o ∧ 𝑚 ∈ ℕ0)
→ 𝑦 = 𝐵) | 
| 15 | 14 | eleq2d 2826 | . . . . . . . . . . . 12
⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑐 ∈ 2o ∧ 𝑚 ∈ ℕ0)
→ (𝑚 ∈ 𝑦 ↔ 𝑚 ∈ 𝐵)) | 
| 16 |  | biidd 262 | . . . . . . . . . . . 12
⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑐 ∈ 2o ∧ 𝑚 ∈ ℕ0)
→ (∅ ∈ 𝑐
↔ ∅ ∈ 𝑐)) | 
| 17 | 13, 15, 16 | cadbi123d 1609 | . . . . . . . . . . 11
⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑐 ∈ 2o ∧ 𝑚 ∈ ℕ0)
→ (cadd(𝑚 ∈ 𝑥, 𝑚 ∈ 𝑦, ∅ ∈ 𝑐) ↔ cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐))) | 
| 18 | 17 | ifbid 4548 | . . . . . . . . . 10
⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑐 ∈ 2o ∧ 𝑚 ∈ ℕ0)
→ if(cadd(𝑚 ∈
𝑥, 𝑚 ∈ 𝑦, ∅ ∈ 𝑐), 1o, ∅) = if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)) | 
| 19 | 18 | mpoeq3dva 7511 | . . . . . . . . 9
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦
if(cadd(𝑚 ∈ 𝑥, 𝑚 ∈ 𝑦, ∅ ∈ 𝑐), 1o, ∅)) = (𝑐 ∈ 2o, 𝑚 ∈ ℕ0
↦ if(cadd(𝑚 ∈
𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅))) | 
| 20 | 19 | seqeq2d 14050 | . . . . . . . 8
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦
if(cadd(𝑚 ∈ 𝑥, 𝑚 ∈ 𝑦, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0
↦ if(𝑛 = 0, ∅,
(𝑛 − 1)))) =
seq0((𝑐 ∈
2o, 𝑚 ∈
ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0
↦ if(𝑛 = 0, ∅,
(𝑛 −
1))))) | 
| 21 |  | sadval.c | . . . . . . . 8
⊢ 𝐶 = seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦
if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0
↦ if(𝑛 = 0, ∅,
(𝑛 −
1)))) | 
| 22 | 20, 21 | eqtr4di 2794 | . . . . . . 7
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦
if(cadd(𝑚 ∈ 𝑥, 𝑚 ∈ 𝑦, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0
↦ if(𝑛 = 0, ∅,
(𝑛 − 1)))) = 𝐶) | 
| 23 | 22 | fveq1d 6907 | . . . . . 6
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦
if(cadd(𝑚 ∈ 𝑥, 𝑚 ∈ 𝑦, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0
↦ if(𝑛 = 0, ∅,
(𝑛 −
1))))‘𝑘) = (𝐶‘𝑘)) | 
| 24 | 23 | eleq2d 2826 | . . . . 5
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∅ ∈ (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0
↦ if(cadd(𝑚 ∈
𝑥, 𝑚 ∈ 𝑦, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0
↦ if(𝑛 = 0, ∅,
(𝑛 −
1))))‘𝑘) ↔
∅ ∈ (𝐶‘𝑘))) | 
| 25 | 9, 11, 24 | hadbi123d 1594 | . . . 4
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (hadd(𝑘 ∈ 𝑥, 𝑘 ∈ 𝑦, ∅ ∈ (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦
if(cadd(𝑚 ∈ 𝑥, 𝑚 ∈ 𝑦, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0
↦ if(𝑛 = 0, ∅,
(𝑛 −
1))))‘𝑘)) ↔
hadd(𝑘 ∈ 𝐴, 𝑘 ∈ 𝐵, ∅ ∈ (𝐶‘𝑘)))) | 
| 26 | 25 | rabbidv 3443 | . . 3
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → {𝑘 ∈ ℕ0 ∣
hadd(𝑘 ∈ 𝑥, 𝑘 ∈ 𝑦, ∅ ∈ (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦
if(cadd(𝑚 ∈ 𝑥, 𝑚 ∈ 𝑦, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0
↦ if(𝑛 = 0, ∅,
(𝑛 −
1))))‘𝑘))} = {𝑘 ∈ ℕ0
∣ hadd(𝑘 ∈ 𝐴, 𝑘 ∈ 𝐵, ∅ ∈ (𝐶‘𝑘))}) | 
| 27 |  | df-sad 16489 | . . 3
⊢  sadd =
(𝑥 ∈ 𝒫
ℕ0, 𝑦
∈ 𝒫 ℕ0 ↦ {𝑘 ∈ ℕ0 ∣
hadd(𝑘 ∈ 𝑥, 𝑘 ∈ 𝑦, ∅ ∈ (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦
if(cadd(𝑚 ∈ 𝑥, 𝑚 ∈ 𝑦, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0
↦ if(𝑛 = 0, ∅,
(𝑛 −
1))))‘𝑘))}) | 
| 28 | 2 | rabex 5338 | . . 3
⊢ {𝑘 ∈ ℕ0
∣ hadd(𝑘 ∈ 𝐴, 𝑘 ∈ 𝐵, ∅ ∈ (𝐶‘𝑘))} ∈ V | 
| 29 | 26, 27, 28 | ovmpoa 7589 | . 2
⊢ ((𝐴 ∈ 𝒫
ℕ0 ∧ 𝐵
∈ 𝒫 ℕ0) → (𝐴 sadd 𝐵) = {𝑘 ∈ ℕ0 ∣
hadd(𝑘 ∈ 𝐴, 𝑘 ∈ 𝐵, ∅ ∈ (𝐶‘𝑘))}) | 
| 30 | 4, 7, 29 | syl2anc 584 | 1
⊢ (𝜑 → (𝐴 sadd 𝐵) = {𝑘 ∈ ℕ0 ∣
hadd(𝑘 ∈ 𝐴, 𝑘 ∈ 𝐵, ∅ ∈ (𝐶‘𝑘))}) |