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Theorem satf0op 35020
Description: An element of a value of the satisfaction predicate as function over wff codes in the empty model and the empty binary relation expressed as ordered pair. (Contributed by AV, 19-Sep-2023.)
Hypothesis
Ref Expression
satf0op.s 𝑆 = (∅ Sat ∅)
Assertion
Ref Expression
satf0op (𝑁 ∈ ω → (𝑋 ∈ (𝑆𝑁) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑁))))
Distinct variable groups:   𝑥,𝑁   𝑥,𝑆   𝑥,𝑋

Proof of Theorem satf0op
Dummy variables 𝑖 𝑗 𝑦 𝑧 𝑎 𝑏 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6902 . . . 4 (𝑦 = ∅ → (𝑆𝑦) = (𝑆‘∅))
21eleq2d 2815 . . 3 (𝑦 = ∅ → (𝑋 ∈ (𝑆𝑦) ↔ 𝑋 ∈ (𝑆‘∅)))
31eleq2d 2815 . . . . 5 (𝑦 = ∅ → (⟨𝑥, ∅⟩ ∈ (𝑆𝑦) ↔ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅)))
43anbi2d 628 . . . 4 (𝑦 = ∅ → ((𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦)) ↔ (𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅))))
54exbidv 1916 . . 3 (𝑦 = ∅ → (∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦)) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅))))
62, 5bibi12d 344 . 2 (𝑦 = ∅ → ((𝑋 ∈ (𝑆𝑦) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦))) ↔ (𝑋 ∈ (𝑆‘∅) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅)))))
7 fveq2 6902 . . . 4 (𝑦 = 𝑧 → (𝑆𝑦) = (𝑆𝑧))
87eleq2d 2815 . . 3 (𝑦 = 𝑧 → (𝑋 ∈ (𝑆𝑦) ↔ 𝑋 ∈ (𝑆𝑧)))
97eleq2d 2815 . . . . 5 (𝑦 = 𝑧 → (⟨𝑥, ∅⟩ ∈ (𝑆𝑦) ↔ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧)))
109anbi2d 628 . . . 4 (𝑦 = 𝑧 → ((𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦)) ↔ (𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧))))
1110exbidv 1916 . . 3 (𝑦 = 𝑧 → (∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦)) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧))))
128, 11bibi12d 344 . 2 (𝑦 = 𝑧 → ((𝑋 ∈ (𝑆𝑦) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦))) ↔ (𝑋 ∈ (𝑆𝑧) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧)))))
13 fveq2 6902 . . . 4 (𝑦 = suc 𝑧 → (𝑆𝑦) = (𝑆‘suc 𝑧))
1413eleq2d 2815 . . 3 (𝑦 = suc 𝑧 → (𝑋 ∈ (𝑆𝑦) ↔ 𝑋 ∈ (𝑆‘suc 𝑧)))
1513eleq2d 2815 . . . . 5 (𝑦 = suc 𝑧 → (⟨𝑥, ∅⟩ ∈ (𝑆𝑦) ↔ ⟨𝑥, ∅⟩ ∈ (𝑆‘suc 𝑧)))
1615anbi2d 628 . . . 4 (𝑦 = suc 𝑧 → ((𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦)) ↔ (𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘suc 𝑧))))
1716exbidv 1916 . . 3 (𝑦 = suc 𝑧 → (∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦)) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘suc 𝑧))))
1814, 17bibi12d 344 . 2 (𝑦 = suc 𝑧 → ((𝑋 ∈ (𝑆𝑦) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦))) ↔ (𝑋 ∈ (𝑆‘suc 𝑧) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘suc 𝑧)))))
19 fveq2 6902 . . . 4 (𝑦 = 𝑁 → (𝑆𝑦) = (𝑆𝑁))
2019eleq2d 2815 . . 3 (𝑦 = 𝑁 → (𝑋 ∈ (𝑆𝑦) ↔ 𝑋 ∈ (𝑆𝑁)))
2119eleq2d 2815 . . . . 5 (𝑦 = 𝑁 → (⟨𝑥, ∅⟩ ∈ (𝑆𝑦) ↔ ⟨𝑥, ∅⟩ ∈ (𝑆𝑁)))
2221anbi2d 628 . . . 4 (𝑦 = 𝑁 → ((𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦)) ↔ (𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑁))))
2322exbidv 1916 . . 3 (𝑦 = 𝑁 → (∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦)) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑁))))
2420, 23bibi12d 344 . 2 (𝑦 = 𝑁 → ((𝑋 ∈ (𝑆𝑦) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦))) ↔ (𝑋 ∈ (𝑆𝑁) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑁)))))
25 satf0op.s . . . . . 6 𝑆 = (∅ Sat ∅)
2625fveq1i 6903 . . . . 5 (𝑆‘∅) = ((∅ Sat ∅)‘∅)
27 satf00 35017 . . . . 5 ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}
2826, 27eqtri 2756 . . . 4 (𝑆‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}
2928eleq2i 2821 . . 3 (𝑋 ∈ (𝑆‘∅) ↔ 𝑋 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})
30 elopab 5533 . . 3 (𝑋 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))} ↔ ∃𝑥𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))))
31 opeq2 4879 . . . . . . . . . . 11 (𝑦 = ∅ → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ∅⟩)
3231adantr 479 . . . . . . . . . 10 ((𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)) → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ∅⟩)
3332eqeq2d 2739 . . . . . . . . 9 ((𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)) → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨𝑥, ∅⟩))
3433biimpd 228 . . . . . . . 8 ((𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)) → (𝑋 = ⟨𝑥, 𝑦⟩ → 𝑋 = ⟨𝑥, ∅⟩))
3534impcom 406 . . . . . . 7 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) → 𝑋 = ⟨𝑥, ∅⟩)
36 eqidd 2729 . . . . . . . . . 10 (𝑦 = ∅ → ∅ = ∅)
3736anim1i 613 . . . . . . . . 9 ((𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)) → (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)))
3837adantl 480 . . . . . . . 8 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) → (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)))
39 satf00 35017 . . . . . . . . . . 11 ((∅ Sat ∅)‘∅) = {⟨𝑦, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗))}
4026, 39eqtri 2756 . . . . . . . . . 10 (𝑆‘∅) = {⟨𝑦, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗))}
4140eleq2i 2821 . . . . . . . . 9 (⟨𝑥, ∅⟩ ∈ (𝑆‘∅) ↔ ⟨𝑥, ∅⟩ ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗))})
42 vex 3477 . . . . . . . . . 10 𝑥 ∈ V
43 0ex 5311 . . . . . . . . . 10 ∅ ∈ V
44 eqeq1 2732 . . . . . . . . . . 11 (𝑧 = ∅ → (𝑧 = ∅ ↔ ∅ = ∅))
45 eqeq1 2732 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑦 = (𝑖𝑔𝑗) ↔ 𝑥 = (𝑖𝑔𝑗)))
46452rexbidv 3217 . . . . . . . . . . 11 (𝑦 = 𝑥 → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)))
4744, 46bi2anan9r 637 . . . . . . . . . 10 ((𝑦 = 𝑥𝑧 = ∅) → ((𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗)) ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))))
4842, 43, 47opelopaba 5542 . . . . . . . . 9 (⟨𝑥, ∅⟩ ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗))} ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)))
4941, 48bitri 274 . . . . . . . 8 (⟨𝑥, ∅⟩ ∈ (𝑆‘∅) ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)))
5038, 49sylibr 233 . . . . . . 7 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) → ⟨𝑥, ∅⟩ ∈ (𝑆‘∅))
5135, 50jca 510 . . . . . 6 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) → (𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅)))
5251exlimiv 1925 . . . . 5 (∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) → (𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅)))
5331eqeq2d 2739 . . . . . . . 8 (𝑦 = ∅ → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨𝑥, ∅⟩))
54 eqeq1 2732 . . . . . . . . 9 (𝑦 = ∅ → (𝑦 = ∅ ↔ ∅ = ∅))
5554anbi1d 629 . . . . . . . 8 (𝑦 = ∅ → ((𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)) ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))))
5653, 55anbi12d 630 . . . . . . 7 (𝑦 = ∅ → ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) ↔ (𝑋 = ⟨𝑥, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)))))
5743, 56spcev 3595 . . . . . 6 ((𝑋 = ⟨𝑥, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) → ∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))))
5849, 57sylan2b 592 . . . . 5 ((𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅)) → ∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))))
5952, 58impbii 208 . . . 4 (∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) ↔ (𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅)))
6059exbii 1842 . . 3 (∃𝑥𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅)))
6129, 30, 603bitri 296 . 2 (𝑋 ∈ (𝑆‘∅) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅)))
6225satf0suc 35019 . . . . . . 7 (𝑧 ∈ ω → (𝑆‘suc 𝑧) = ((𝑆𝑧) ∪ {⟨𝑎, 𝑏⟩ ∣ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))}))
6362eleq2d 2815 . . . . . 6 (𝑧 ∈ ω → (𝑋 ∈ (𝑆‘suc 𝑧) ↔ 𝑋 ∈ ((𝑆𝑧) ∪ {⟨𝑎, 𝑏⟩ ∣ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))})))
64 elun 4149 . . . . . . 7 (𝑋 ∈ ((𝑆𝑧) ∪ {⟨𝑎, 𝑏⟩ ∣ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))}) ↔ (𝑋 ∈ (𝑆𝑧) ∨ 𝑋 ∈ {⟨𝑎, 𝑏⟩ ∣ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))}))
6564a1i 11 . . . . . 6 (𝑧 ∈ ω → (𝑋 ∈ ((𝑆𝑧) ∪ {⟨𝑎, 𝑏⟩ ∣ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))}) ↔ (𝑋 ∈ (𝑆𝑧) ∨ 𝑋 ∈ {⟨𝑎, 𝑏⟩ ∣ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))})))
66 elopab 5533 . . . . . . . 8 (𝑋 ∈ {⟨𝑎, 𝑏⟩ ∣ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))} ↔ ∃𝑎𝑏(𝑋 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))))
6766a1i 11 . . . . . . 7 (𝑧 ∈ ω → (𝑋 ∈ {⟨𝑎, 𝑏⟩ ∣ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))} ↔ ∃𝑎𝑏(𝑋 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢))))))
6867orbi2d 913 . . . . . 6 (𝑧 ∈ ω → ((𝑋 ∈ (𝑆𝑧) ∨ 𝑋 ∈ {⟨𝑎, 𝑏⟩ ∣ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))}) ↔ (𝑋 ∈ (𝑆𝑧) ∨ ∃𝑎𝑏(𝑋 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))))))
6963, 65, 683bitrd 304 . . . . 5 (𝑧 ∈ ω → (𝑋 ∈ (𝑆‘suc 𝑧) ↔ (𝑋 ∈ (𝑆𝑧) ∨ ∃𝑎𝑏(𝑋 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))))))
7069adantr 479 . . . 4 ((𝑧 ∈ ω ∧ (𝑋 ∈ (𝑆𝑧) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧)))) → (𝑋 ∈ (𝑆‘suc 𝑧) ↔ (𝑋 ∈ (𝑆𝑧) ∨ ∃𝑎𝑏(𝑋 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))))))
71 simpr 483 . . . . . 6 ((𝑧 ∈ ω ∧ (𝑋 ∈ (𝑆𝑧) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧)))) → (𝑋 ∈ (𝑆𝑧) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧))))
72 opeq2 4879 . . . . . . . . . . . . . . . . 17 (𝑏 = ∅ → ⟨𝑎, 𝑏⟩ = ⟨𝑎, ∅⟩)
7372eqeq2d 2739 . . . . . . . . . . . . . . . 16 (𝑏 = ∅ → (𝑋 = ⟨𝑎, 𝑏⟩ ↔ 𝑋 = ⟨𝑎, ∅⟩))
7473biimpd 228 . . . . . . . . . . . . . . 15 (𝑏 = ∅ → (𝑋 = ⟨𝑎, 𝑏⟩ → 𝑋 = ⟨𝑎, ∅⟩))
7574adantr 479 . . . . . . . . . . . . . 14 ((𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢))) → (𝑋 = ⟨𝑎, 𝑏⟩ → 𝑋 = ⟨𝑎, ∅⟩))
7675impcom 406 . . . . . . . . . . . . 13 ((𝑋 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))) → 𝑋 = ⟨𝑎, ∅⟩)
77 eqidd 2729 . . . . . . . . . . . . . 14 ((𝑋 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))) → ∅ = ∅)
78 simpr 483 . . . . . . . . . . . . . . 15 ((𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢))) → ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))
7978adantl 480 . . . . . . . . . . . . . 14 ((𝑋 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))) → ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))
8077, 79jca 510 . . . . . . . . . . . . 13 ((𝑋 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))) → (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢))))
8176, 80jca 510 . . . . . . . . . . . 12 ((𝑋 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))) → (𝑋 = ⟨𝑎, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))))
8281exlimiv 1925 . . . . . . . . . . 11 (∃𝑏(𝑋 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))) → (𝑋 = ⟨𝑎, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))))
83 eqeq1 2732 . . . . . . . . . . . . . 14 (𝑏 = ∅ → (𝑏 = ∅ ↔ ∅ = ∅))
8483anbi1d 629 . . . . . . . . . . . . 13 (𝑏 = ∅ → ((𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢))) ↔ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))))
8573, 84anbi12d 630 . . . . . . . . . . . 12 (𝑏 = ∅ → ((𝑋 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))) ↔ (𝑋 = ⟨𝑎, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢))))))
8643, 85spcev 3595 . . . . . . . . . . 11 ((𝑋 = ⟨𝑎, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))) → ∃𝑏(𝑋 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))))
8782, 86impbii 208 . . . . . . . . . 10 (∃𝑏(𝑋 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))) ↔ (𝑋 = ⟨𝑎, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))))
8887exbii 1842 . . . . . . . . 9 (∃𝑎𝑏(𝑋 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))) ↔ ∃𝑎(𝑋 = ⟨𝑎, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))))
8988a1i 11 . . . . . . . 8 (𝑧 ∈ ω → (∃𝑎𝑏(𝑋 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))) ↔ ∃𝑎(𝑋 = ⟨𝑎, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢))))))
90 opeq1 4878 . . . . . . . . . . 11 (𝑥 = 𝑎 → ⟨𝑥, ∅⟩ = ⟨𝑎, ∅⟩)
9190eqeq2d 2739 . . . . . . . . . 10 (𝑥 = 𝑎 → (𝑋 = ⟨𝑥, ∅⟩ ↔ 𝑋 = ⟨𝑎, ∅⟩))
92 eqeq1 2732 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ 𝑎 = ((1st𝑢)⊼𝑔(1st𝑣))))
9392rexbidv 3176 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ ∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣))))
94 eqeq1 2732 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (𝑥 = ∀𝑔𝑖(1st𝑢) ↔ 𝑎 = ∀𝑔𝑖(1st𝑢)))
9594rexbidv 3176 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢) ↔ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))
9693, 95orbi12d 916 . . . . . . . . . . . 12 (𝑥 = 𝑎 → ((∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) ↔ (∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢))))
9796rexbidv 3176 . . . . . . . . . . 11 (𝑥 = 𝑎 → (∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) ↔ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢))))
9897anbi2d 628 . . . . . . . . . 10 (𝑥 = 𝑎 → ((∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))) ↔ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))))
9991, 98anbi12d 630 . . . . . . . . 9 (𝑥 = 𝑎 → ((𝑋 = ⟨𝑥, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))) ↔ (𝑋 = ⟨𝑎, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢))))))
10099cbvexvw 2032 . . . . . . . 8 (∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))) ↔ ∃𝑎(𝑋 = ⟨𝑎, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))))
10189, 100bitr4di 288 . . . . . . 7 (𝑧 ∈ ω → (∃𝑎𝑏(𝑋 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))))
102101adantr 479 . . . . . 6 ((𝑧 ∈ ω ∧ (𝑋 ∈ (𝑆𝑧) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧)))) → (∃𝑎𝑏(𝑋 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))))
10371, 102orbi12d 916 . . . . 5 ((𝑧 ∈ ω ∧ (𝑋 ∈ (𝑆𝑧) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧)))) → ((𝑋 ∈ (𝑆𝑧) ∨ ∃𝑎𝑏(𝑋 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢))))) ↔ (∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧)) ∨ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))))))
104 19.43 1877 . . . . . 6 (∃𝑥((𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧)) ∨ (𝑋 = ⟨𝑥, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))) ↔ (∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧)) ∨ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))))
105 andi 1005 . . . . . . . 8 ((𝑋 = ⟨𝑥, ∅⟩ ∧ (⟨𝑥, ∅⟩ ∈ (𝑆𝑧) ∨ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))) ↔ ((𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧)) ∨ (𝑋 = ⟨𝑥, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))))
106105bicomi 223 . . . . . . 7 (((𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧)) ∨ (𝑋 = ⟨𝑥, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))) ↔ (𝑋 = ⟨𝑥, ∅⟩ ∧ (⟨𝑥, ∅⟩ ∈ (𝑆𝑧) ∨ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))))
107106exbii 1842 . . . . . 6 (∃𝑥((𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧)) ∨ (𝑋 = ⟨𝑥, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ (⟨𝑥, ∅⟩ ∈ (𝑆𝑧) ∨ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))))
108104, 107bitr3i 276 . . . . 5 ((∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧)) ∨ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ (⟨𝑥, ∅⟩ ∈ (𝑆𝑧) ∨ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))))
109103, 108bitrdi 286 . . . 4 ((𝑧 ∈ ω ∧ (𝑋 ∈ (𝑆𝑧) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧)))) → ((𝑋 ∈ (𝑆𝑧) ∨ ∃𝑎𝑏(𝑋 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢))))) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ (⟨𝑥, ∅⟩ ∈ (𝑆𝑧) ∨ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))))))
11062eleq2d 2815 . . . . . . . . 9 (𝑧 ∈ ω → (⟨𝑥, ∅⟩ ∈ (𝑆‘suc 𝑧) ↔ ⟨𝑥, ∅⟩ ∈ ((𝑆𝑧) ∪ {⟨𝑎, 𝑏⟩ ∣ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))})))
111 elun 4149 . . . . . . . . . 10 (⟨𝑥, ∅⟩ ∈ ((𝑆𝑧) ∪ {⟨𝑎, 𝑏⟩ ∣ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))}) ↔ (⟨𝑥, ∅⟩ ∈ (𝑆𝑧) ∨ ⟨𝑥, ∅⟩ ∈ {⟨𝑎, 𝑏⟩ ∣ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))}))
112 eqeq1 2732 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → (𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
113112rexbidv 3176 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → (∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ ∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
114 eqeq1 2732 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → (𝑎 = ∀𝑔𝑖(1st𝑢) ↔ 𝑥 = ∀𝑔𝑖(1st𝑢)))
115114rexbidv 3176 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → (∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢) ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))
116113, 115orbi12d 916 . . . . . . . . . . . . . 14 (𝑎 = 𝑥 → ((∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)) ↔ (∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
117116rexbidv 3176 . . . . . . . . . . . . 13 (𝑎 = 𝑥 → (∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)) ↔ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
11883, 117bi2anan9r 637 . . . . . . . . . . . 12 ((𝑎 = 𝑥𝑏 = ∅) → ((𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢))) ↔ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))))
11942, 43, 118opelopaba 5542 . . . . . . . . . . 11 (⟨𝑥, ∅⟩ ∈ {⟨𝑎, 𝑏⟩ ∣ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))} ↔ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
120119orbi2i 910 . . . . . . . . . 10 ((⟨𝑥, ∅⟩ ∈ (𝑆𝑧) ∨ ⟨𝑥, ∅⟩ ∈ {⟨𝑎, 𝑏⟩ ∣ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))}) ↔ (⟨𝑥, ∅⟩ ∈ (𝑆𝑧) ∨ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))))
121111, 120bitri 274 . . . . . . . . 9 (⟨𝑥, ∅⟩ ∈ ((𝑆𝑧) ∪ {⟨𝑎, 𝑏⟩ ∣ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))}) ↔ (⟨𝑥, ∅⟩ ∈ (𝑆𝑧) ∨ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))))
122110, 121bitrdi 286 . . . . . . . 8 (𝑧 ∈ ω → (⟨𝑥, ∅⟩ ∈ (𝑆‘suc 𝑧) ↔ (⟨𝑥, ∅⟩ ∈ (𝑆𝑧) ∨ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))))
123122anbi2d 628 . . . . . . 7 (𝑧 ∈ ω → ((𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘suc 𝑧)) ↔ (𝑋 = ⟨𝑥, ∅⟩ ∧ (⟨𝑥, ∅⟩ ∈ (𝑆𝑧) ∨ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))))))
124123exbidv 1916 . . . . . 6 (𝑧 ∈ ω → (∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘suc 𝑧)) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ (⟨𝑥, ∅⟩ ∈ (𝑆𝑧) ∨ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))))))
125124bicomd 222 . . . . 5 (𝑧 ∈ ω → (∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ (⟨𝑥, ∅⟩ ∈ (𝑆𝑧) ∨ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘suc 𝑧))))
126125adantr 479 . . . 4 ((𝑧 ∈ ω ∧ (𝑋 ∈ (𝑆𝑧) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧)))) → (∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ (⟨𝑥, ∅⟩ ∈ (𝑆𝑧) ∨ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘suc 𝑧))))
12770, 109, 1263bitrd 304 . . 3 ((𝑧 ∈ ω ∧ (𝑋 ∈ (𝑆𝑧) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧)))) → (𝑋 ∈ (𝑆‘suc 𝑧) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘suc 𝑧))))
128127ex 411 . 2 (𝑧 ∈ ω → ((𝑋 ∈ (𝑆𝑧) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧))) → (𝑋 ∈ (𝑆‘suc 𝑧) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘suc 𝑧)))))
1296, 12, 18, 24, 61, 128finds 7910 1 (𝑁 ∈ ω → (𝑋 ∈ (𝑆𝑁) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑁))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wo 845   = wceq 1533  wex 1773  wcel 2098  wrex 3067  cun 3947  c0 4326  cop 4638  {copab 5214  suc csuc 6376  cfv 6553  (class class class)co 7426  ωcom 7876  1st c1st 7997  𝑔cgoe 34976  𝑔cgna 34977  𝑔cgol 34978   Sat csat 34979
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-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9672
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-map 8853  df-goel 34983  df-sat 34986
This theorem is referenced by:  fmlasuc  35029
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