| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | elnn 7898 | . . . 4
⊢ ((𝐵 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝐵 ∈ ω) | 
| 2 | 1 | ancoms 458 | . . 3
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ ω) | 
| 3 |  | nnord 7895 | . . . . . . 7
⊢ (𝐴 ∈ ω → Ord 𝐴) | 
| 4 |  | ordsucss 7838 | . . . . . . 7
⊢ (Ord
𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴)) | 
| 5 | 3, 4 | syl 17 | . . . . . 6
⊢ (𝐴 ∈ ω → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴)) | 
| 6 | 5 | adantr 480 | . . . . 5
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴)) | 
| 7 |  | peano2b 7904 | . . . . . 6
⊢ (𝐵 ∈ ω ↔ suc 𝐵 ∈
ω) | 
| 8 |  | fveq2 6906 | . . . . . . . . . 10
⊢ (𝑣 = suc 𝐵 → (𝐹‘𝑣) = (𝐹‘suc 𝐵)) | 
| 9 | 8 | psseq2d 4096 | . . . . . . . . 9
⊢ (𝑣 = suc 𝐵 → ((𝐹‘𝐵) ⊊ (𝐹‘𝑣) ↔ (𝐹‘𝐵) ⊊ (𝐹‘suc 𝐵))) | 
| 10 | 9 | imbi2d 340 | . . . . . . . 8
⊢ (𝑣 = suc 𝐵 → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝑣)) ↔ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝐵)))) | 
| 11 |  | fveq2 6906 | . . . . . . . . . 10
⊢ (𝑣 = 𝑢 → (𝐹‘𝑣) = (𝐹‘𝑢)) | 
| 12 | 11 | psseq2d 4096 | . . . . . . . . 9
⊢ (𝑣 = 𝑢 → ((𝐹‘𝐵) ⊊ (𝐹‘𝑣) ↔ (𝐹‘𝐵) ⊊ (𝐹‘𝑢))) | 
| 13 | 12 | imbi2d 340 | . . . . . . . 8
⊢ (𝑣 = 𝑢 → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝑣)) ↔ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝑢)))) | 
| 14 |  | fveq2 6906 | . . . . . . . . . 10
⊢ (𝑣 = suc 𝑢 → (𝐹‘𝑣) = (𝐹‘suc 𝑢)) | 
| 15 | 14 | psseq2d 4096 | . . . . . . . . 9
⊢ (𝑣 = suc 𝑢 → ((𝐹‘𝐵) ⊊ (𝐹‘𝑣) ↔ (𝐹‘𝐵) ⊊ (𝐹‘suc 𝑢))) | 
| 16 | 15 | imbi2d 340 | . . . . . . . 8
⊢ (𝑣 = suc 𝑢 → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝑣)) ↔ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝑢)))) | 
| 17 |  | fveq2 6906 | . . . . . . . . . 10
⊢ (𝑣 = 𝐴 → (𝐹‘𝑣) = (𝐹‘𝐴)) | 
| 18 | 17 | psseq2d 4096 | . . . . . . . . 9
⊢ (𝑣 = 𝐴 → ((𝐹‘𝐵) ⊊ (𝐹‘𝑣) ↔ (𝐹‘𝐵) ⊊ (𝐹‘𝐴))) | 
| 19 | 18 | imbi2d 340 | . . . . . . . 8
⊢ (𝑣 = 𝐴 → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝑣)) ↔ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴)))) | 
| 20 |  | inf3lem.1 | . . . . . . . . . . 11
⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) | 
| 21 |  | inf3lem.2 | . . . . . . . . . . 11
⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω) | 
| 22 |  | inf3lem.4 | . . . . . . . . . . 11
⊢ 𝐵 ∈ V | 
| 23 | 20, 21, 22, 22 | inf3lem4 9671 | . . . . . . . . . 10
⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥)
→ (𝐵 ∈ ω
→ (𝐹‘𝐵) ⊊ (𝐹‘suc 𝐵))) | 
| 24 | 23 | com12 32 | . . . . . . . . 9
⊢ (𝐵 ∈ ω → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥)
→ (𝐹‘𝐵) ⊊ (𝐹‘suc 𝐵))) | 
| 25 | 7, 24 | sylbir 235 | . . . . . . . 8
⊢ (suc
𝐵 ∈ ω →
((𝑥 ≠ ∅ ∧
𝑥 ⊆ ∪ 𝑥)
→ (𝐹‘𝐵) ⊊ (𝐹‘suc 𝐵))) | 
| 26 |  | vex 3484 | . . . . . . . . . . . 12
⊢ 𝑢 ∈ V | 
| 27 | 20, 21, 26, 22 | inf3lem4 9671 | . . . . . . . . . . 11
⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥)
→ (𝑢 ∈ ω
→ (𝐹‘𝑢) ⊊ (𝐹‘suc 𝑢))) | 
| 28 |  | psstr 4107 | . . . . . . . . . . . 12
⊢ (((𝐹‘𝐵) ⊊ (𝐹‘𝑢) ∧ (𝐹‘𝑢) ⊊ (𝐹‘suc 𝑢)) → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝑢)) | 
| 29 | 28 | expcom 413 | . . . . . . . . . . 11
⊢ ((𝐹‘𝑢) ⊊ (𝐹‘suc 𝑢) → ((𝐹‘𝐵) ⊊ (𝐹‘𝑢) → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝑢))) | 
| 30 | 27, 29 | syl6com 37 | . . . . . . . . . 10
⊢ (𝑢 ∈ ω → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥)
→ ((𝐹‘𝐵) ⊊ (𝐹‘𝑢) → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝑢)))) | 
| 31 | 30 | a2d 29 | . . . . . . . . 9
⊢ (𝑢 ∈ ω → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥)
→ (𝐹‘𝐵) ⊊ (𝐹‘𝑢)) → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝑢)))) | 
| 32 | 31 | ad2antrr 726 | . . . . . . . 8
⊢ (((𝑢 ∈ ω ∧ suc 𝐵 ∈ ω) ∧ suc 𝐵 ⊆ 𝑢) → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝑢)) → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝑢)))) | 
| 33 | 10, 13, 16, 19, 25, 32 | findsg 7919 | . . . . . . 7
⊢ (((𝐴 ∈ ω ∧ suc 𝐵 ∈ ω) ∧ suc 𝐵 ⊆ 𝐴) → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴))) | 
| 34 | 33 | ex 412 | . . . . . 6
⊢ ((𝐴 ∈ ω ∧ suc 𝐵 ∈ ω) → (suc
𝐵 ⊆ 𝐴 → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴)))) | 
| 35 | 7, 34 | sylan2b 594 | . . . . 5
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc
𝐵 ⊆ 𝐴 → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴)))) | 
| 36 | 6, 35 | syld 47 | . . . 4
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ∈ 𝐴 → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴)))) | 
| 37 | 36 | impancom 451 | . . 3
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → (𝐵 ∈ ω → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴)))) | 
| 38 | 2, 37 | mpd 15 | . 2
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴))) | 
| 39 | 38 | com12 32 | 1
⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥)
→ ((𝐴 ∈ ω
∧ 𝐵 ∈ 𝐴) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴))) |