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Theorem satf0op 33971
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 6842 . . . 4 (𝑦 = ∅ → (𝑆𝑦) = (𝑆‘∅))
21eleq2d 2823 . . 3 (𝑦 = ∅ → (𝑋 ∈ (𝑆𝑦) ↔ 𝑋 ∈ (𝑆‘∅)))
31eleq2d 2823 . . . . 5 (𝑦 = ∅ → (⟨𝑥, ∅⟩ ∈ (𝑆𝑦) ↔ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅)))
43anbi2d 629 . . . 4 (𝑦 = ∅ → ((𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦)) ↔ (𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅))))
54exbidv 1924 . . 3 (𝑦 = ∅ → (∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦)) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅))))
62, 5bibi12d 345 . 2 (𝑦 = ∅ → ((𝑋 ∈ (𝑆𝑦) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦))) ↔ (𝑋 ∈ (𝑆‘∅) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅)))))
7 fveq2 6842 . . . 4 (𝑦 = 𝑧 → (𝑆𝑦) = (𝑆𝑧))
87eleq2d 2823 . . 3 (𝑦 = 𝑧 → (𝑋 ∈ (𝑆𝑦) ↔ 𝑋 ∈ (𝑆𝑧)))
97eleq2d 2823 . . . . 5 (𝑦 = 𝑧 → (⟨𝑥, ∅⟩ ∈ (𝑆𝑦) ↔ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧)))
109anbi2d 629 . . . 4 (𝑦 = 𝑧 → ((𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦)) ↔ (𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧))))
1110exbidv 1924 . . 3 (𝑦 = 𝑧 → (∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦)) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧))))
128, 11bibi12d 345 . 2 (𝑦 = 𝑧 → ((𝑋 ∈ (𝑆𝑦) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦))) ↔ (𝑋 ∈ (𝑆𝑧) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧)))))
13 fveq2 6842 . . . 4 (𝑦 = suc 𝑧 → (𝑆𝑦) = (𝑆‘suc 𝑧))
1413eleq2d 2823 . . 3 (𝑦 = suc 𝑧 → (𝑋 ∈ (𝑆𝑦) ↔ 𝑋 ∈ (𝑆‘suc 𝑧)))
1513eleq2d 2823 . . . . 5 (𝑦 = suc 𝑧 → (⟨𝑥, ∅⟩ ∈ (𝑆𝑦) ↔ ⟨𝑥, ∅⟩ ∈ (𝑆‘suc 𝑧)))
1615anbi2d 629 . . . 4 (𝑦 = suc 𝑧 → ((𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦)) ↔ (𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘suc 𝑧))))
1716exbidv 1924 . . 3 (𝑦 = suc 𝑧 → (∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦)) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘suc 𝑧))))
1814, 17bibi12d 345 . 2 (𝑦 = suc 𝑧 → ((𝑋 ∈ (𝑆𝑦) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦))) ↔ (𝑋 ∈ (𝑆‘suc 𝑧) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘suc 𝑧)))))
19 fveq2 6842 . . . 4 (𝑦 = 𝑁 → (𝑆𝑦) = (𝑆𝑁))
2019eleq2d 2823 . . 3 (𝑦 = 𝑁 → (𝑋 ∈ (𝑆𝑦) ↔ 𝑋 ∈ (𝑆𝑁)))
2119eleq2d 2823 . . . . 5 (𝑦 = 𝑁 → (⟨𝑥, ∅⟩ ∈ (𝑆𝑦) ↔ ⟨𝑥, ∅⟩ ∈ (𝑆𝑁)))
2221anbi2d 629 . . . 4 (𝑦 = 𝑁 → ((𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦)) ↔ (𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑁))))
2322exbidv 1924 . . 3 (𝑦 = 𝑁 → (∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦)) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑁))))
2420, 23bibi12d 345 . 2 (𝑦 = 𝑁 → ((𝑋 ∈ (𝑆𝑦) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦))) ↔ (𝑋 ∈ (𝑆𝑁) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑁)))))
25 satf0op.s . . . . . 6 𝑆 = (∅ Sat ∅)
2625fveq1i 6843 . . . . 5 (𝑆‘∅) = ((∅ Sat ∅)‘∅)
27 satf00 33968 . . . . 5 ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}
2826, 27eqtri 2764 . . . 4 (𝑆‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}
2928eleq2i 2829 . . 3 (𝑋 ∈ (𝑆‘∅) ↔ 𝑋 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})
30 elopab 5484 . . 3 (𝑋 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))} ↔ ∃𝑥𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))))
31 opeq2 4831 . . . . . . . . . . 11 (𝑦 = ∅ → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ∅⟩)
3231adantr 481 . . . . . . . . . 10 ((𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)) → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ∅⟩)
3332eqeq2d 2747 . . . . . . . . 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 408 . . . . . . 7 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) → 𝑋 = ⟨𝑥, ∅⟩)
36 eqidd 2737 . . . . . . . . . 10 (𝑦 = ∅ → ∅ = ∅)
3736anim1i 615 . . . . . . . . 9 ((𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)) → (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)))
3837adantl 482 . . . . . . . 8 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) → (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)))
39 satf00 33968 . . . . . . . . . . 11 ((∅ Sat ∅)‘∅) = {⟨𝑦, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗))}
4026, 39eqtri 2764 . . . . . . . . . 10 (𝑆‘∅) = {⟨𝑦, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗))}
4140eleq2i 2829 . . . . . . . . 9 (⟨𝑥, ∅⟩ ∈ (𝑆‘∅) ↔ ⟨𝑥, ∅⟩ ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗))})
42 vex 3449 . . . . . . . . . 10 𝑥 ∈ V
43 0ex 5264 . . . . . . . . . 10 ∅ ∈ V
44 eqeq1 2740 . . . . . . . . . . 11 (𝑧 = ∅ → (𝑧 = ∅ ↔ ∅ = ∅))
45 eqeq1 2740 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑦 = (𝑖𝑔𝑗) ↔ 𝑥 = (𝑖𝑔𝑗)))
46452rexbidv 3213 . . . . . . . . . . 11 (𝑦 = 𝑥 → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)))
4744, 46bi2anan9r 638 . . . . . . . . . 10 ((𝑦 = 𝑥𝑧 = ∅) → ((𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗)) ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))))
4842, 43, 47opelopaba 5493 . . . . . . . . 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 512 . . . . . 6 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) → (𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅)))
5251exlimiv 1933 . . . . 5 (∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) → (𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅)))
5331eqeq2d 2747 . . . . . . . 8 (𝑦 = ∅ → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨𝑥, ∅⟩))
54 eqeq1 2740 . . . . . . . . 9 (𝑦 = ∅ → (𝑦 = ∅ ↔ ∅ = ∅))
5554anbi1d 630 . . . . . . . 8 (𝑦 = ∅ → ((𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)) ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))))
5653, 55anbi12d 631 . . . . . . 7 (𝑦 = ∅ → ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) ↔ (𝑋 = ⟨𝑥, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)))))
5743, 56spcev 3565 . . . . . 6 ((𝑋 = ⟨𝑥, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) → ∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))))
5849, 57sylan2b 594 . . . . 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 1850 . . 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 33970 . . . . . . 7 (𝑧 ∈ ω → (𝑆‘suc 𝑧) = ((𝑆𝑧) ∪ {⟨𝑎, 𝑏⟩ ∣ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))}))
6362eleq2d 2823 . . . . . 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 4108 . . . . . . 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 5484 . . . . . . . 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 914 . . . . . 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 481 . . . 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 485 . . . . . 6 ((𝑧 ∈ ω ∧ (𝑋 ∈ (𝑆𝑧) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧)))) → (𝑋 ∈ (𝑆𝑧) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧))))
72 opeq2 4831 . . . . . . . . . . . . . . . . 17 (𝑏 = ∅ → ⟨𝑎, 𝑏⟩ = ⟨𝑎, ∅⟩)
7372eqeq2d 2747 . . . . . . . . . . . . . . . 16 (𝑏 = ∅ → (𝑋 = ⟨𝑎, 𝑏⟩ ↔ 𝑋 = ⟨𝑎, ∅⟩))
7473biimpd 228 . . . . . . . . . . . . . . 15 (𝑏 = ∅ → (𝑋 = ⟨𝑎, 𝑏⟩ → 𝑋 = ⟨𝑎, ∅⟩))
7574adantr 481 . . . . . . . . . . . . . 14 ((𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢))) → (𝑋 = ⟨𝑎, 𝑏⟩ → 𝑋 = ⟨𝑎, ∅⟩))
7675impcom 408 . . . . . . . . . . . . 13 ((𝑋 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))) → 𝑋 = ⟨𝑎, ∅⟩)
77 eqidd 2737 . . . . . . . . . . . . . 14 ((𝑋 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))) → ∅ = ∅)
78 simpr 485 . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . 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 512 . . . . . . . . . . . . 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 512 . . . . . . . . . . . 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 1933 . . . . . . . . . . 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 2740 . . . . . . . . . . . . . 14 (𝑏 = ∅ → (𝑏 = ∅ ↔ ∅ = ∅))
8483anbi1d 630 . . . . . . . . . . . . 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 631 . . . . . . . . . . . 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 3565 . . . . . . . . . . 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 1850 . . . . . . . . 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 4830 . . . . . . . . . . 11 (𝑥 = 𝑎 → ⟨𝑥, ∅⟩ = ⟨𝑎, ∅⟩)
9190eqeq2d 2747 . . . . . . . . . 10 (𝑥 = 𝑎 → (𝑋 = ⟨𝑥, ∅⟩ ↔ 𝑋 = ⟨𝑎, ∅⟩))
92 eqeq1 2740 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ 𝑎 = ((1st𝑢)⊼𝑔(1st𝑣))))
9392rexbidv 3175 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ ∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣))))
94 eqeq1 2740 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (𝑥 = ∀𝑔𝑖(1st𝑢) ↔ 𝑎 = ∀𝑔𝑖(1st𝑢)))
9594rexbidv 3175 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢) ↔ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))
9693, 95orbi12d 917 . . . . . . . . . . . 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 3175 . . . . . . . . . . 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 629 . . . . . . . . . 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 631 . . . . . . . . 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 2040 . . . . . . . 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 481 . . . . . 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 917 . . . . 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 1885 . . . . . 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 1006 . . . . . . . 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 1850 . . . . . 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 2823 . . . . . . . . 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 4108 . . . . . . . . . 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 2740 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → (𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
113112rexbidv 3175 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → (∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ ∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
114 eqeq1 2740 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → (𝑎 = ∀𝑔𝑖(1st𝑢) ↔ 𝑥 = ∀𝑔𝑖(1st𝑢)))
115114rexbidv 3175 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → (∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢) ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))
116113, 115orbi12d 917 . . . . . . . . . . . . . 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 3175 . . . . . . . . . . . . 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 638 . . . . . . . . . . . 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 5493 . . . . . . . . . . 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 911 . . . . . . . . . 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 629 . . . . . . 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 1924 . . . . . 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 481 . . . 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 413 . 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 7835 1 (𝑁 ∈ ω → (𝑋 ∈ (𝑆𝑁) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑁))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wex 1781  wcel 2106  wrex 3073  cun 3908  c0 4282  cop 4592  {copab 5167  suc csuc 6319  cfv 6496  (class class class)co 7357  ωcom 7802  1st c1st 7919  𝑔cgoe 33927  𝑔cgna 33928  𝑔cgol 33929   Sat csat 33930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-map 8767  df-goel 33934  df-sat 33937
This theorem is referenced by:  fmlasuc  33980
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