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Theorem satf0op 35362
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 6907 . . . 4 (𝑦 = ∅ → (𝑆𝑦) = (𝑆‘∅))
21eleq2d 2825 . . 3 (𝑦 = ∅ → (𝑋 ∈ (𝑆𝑦) ↔ 𝑋 ∈ (𝑆‘∅)))
31eleq2d 2825 . . . . 5 (𝑦 = ∅ → (⟨𝑥, ∅⟩ ∈ (𝑆𝑦) ↔ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅)))
43anbi2d 630 . . . 4 (𝑦 = ∅ → ((𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦)) ↔ (𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅))))
54exbidv 1919 . . 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 6907 . . . 4 (𝑦 = 𝑧 → (𝑆𝑦) = (𝑆𝑧))
87eleq2d 2825 . . 3 (𝑦 = 𝑧 → (𝑋 ∈ (𝑆𝑦) ↔ 𝑋 ∈ (𝑆𝑧)))
97eleq2d 2825 . . . . 5 (𝑦 = 𝑧 → (⟨𝑥, ∅⟩ ∈ (𝑆𝑦) ↔ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧)))
109anbi2d 630 . . . 4 (𝑦 = 𝑧 → ((𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦)) ↔ (𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧))))
1110exbidv 1919 . . 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 6907 . . . 4 (𝑦 = suc 𝑧 → (𝑆𝑦) = (𝑆‘suc 𝑧))
1413eleq2d 2825 . . 3 (𝑦 = suc 𝑧 → (𝑋 ∈ (𝑆𝑦) ↔ 𝑋 ∈ (𝑆‘suc 𝑧)))
1513eleq2d 2825 . . . . 5 (𝑦 = suc 𝑧 → (⟨𝑥, ∅⟩ ∈ (𝑆𝑦) ↔ ⟨𝑥, ∅⟩ ∈ (𝑆‘suc 𝑧)))
1615anbi2d 630 . . . 4 (𝑦 = suc 𝑧 → ((𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦)) ↔ (𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘suc 𝑧))))
1716exbidv 1919 . . 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 6907 . . . 4 (𝑦 = 𝑁 → (𝑆𝑦) = (𝑆𝑁))
2019eleq2d 2825 . . 3 (𝑦 = 𝑁 → (𝑋 ∈ (𝑆𝑦) ↔ 𝑋 ∈ (𝑆𝑁)))
2119eleq2d 2825 . . . . 5 (𝑦 = 𝑁 → (⟨𝑥, ∅⟩ ∈ (𝑆𝑦) ↔ ⟨𝑥, ∅⟩ ∈ (𝑆𝑁)))
2221anbi2d 630 . . . 4 (𝑦 = 𝑁 → ((𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦)) ↔ (𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑁))))
2322exbidv 1919 . . 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 6908 . . . . 5 (𝑆‘∅) = ((∅ Sat ∅)‘∅)
27 satf00 35359 . . . . 5 ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}
2826, 27eqtri 2763 . . . 4 (𝑆‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}
2928eleq2i 2831 . . 3 (𝑋 ∈ (𝑆‘∅) ↔ 𝑋 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})
30 elopab 5537 . . 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 480 . . . . . . . . . 10 ((𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)) → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ∅⟩)
3332eqeq2d 2746 . . . . . . . . 9 ((𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)) → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨𝑥, ∅⟩))
3433biimpd 229 . . . . . . . 8 ((𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)) → (𝑋 = ⟨𝑥, 𝑦⟩ → 𝑋 = ⟨𝑥, ∅⟩))
3534impcom 407 . . . . . . 7 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) → 𝑋 = ⟨𝑥, ∅⟩)
36 eqidd 2736 . . . . . . . . . 10 (𝑦 = ∅ → ∅ = ∅)
3736anim1i 615 . . . . . . . . 9 ((𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)) → (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)))
3837adantl 481 . . . . . . . 8 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) → (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)))
39 satf00 35359 . . . . . . . . . . 11 ((∅ Sat ∅)‘∅) = {⟨𝑦, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗))}
4026, 39eqtri 2763 . . . . . . . . . 10 (𝑆‘∅) = {⟨𝑦, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗))}
4140eleq2i 2831 . . . . . . . . 9 (⟨𝑥, ∅⟩ ∈ (𝑆‘∅) ↔ ⟨𝑥, ∅⟩ ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗))})
42 vex 3482 . . . . . . . . . 10 𝑥 ∈ V
43 0ex 5313 . . . . . . . . . 10 ∅ ∈ V
44 eqeq1 2739 . . . . . . . . . . 11 (𝑧 = ∅ → (𝑧 = ∅ ↔ ∅ = ∅))
45 eqeq1 2739 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑦 = (𝑖𝑔𝑗) ↔ 𝑥 = (𝑖𝑔𝑗)))
46452rexbidv 3220 . . . . . . . . . . 11 (𝑦 = 𝑥 → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)))
4744, 46bi2anan9r 639 . . . . . . . . . 10 ((𝑦 = 𝑥𝑧 = ∅) → ((𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗)) ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))))
4842, 43, 47opelopaba 5546 . . . . . . . . 9 (⟨𝑥, ∅⟩ ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗))} ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)))
4941, 48bitri 275 . . . . . . . 8 (⟨𝑥, ∅⟩ ∈ (𝑆‘∅) ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)))
5038, 49sylibr 234 . . . . . . 7 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) → ⟨𝑥, ∅⟩ ∈ (𝑆‘∅))
5135, 50jca 511 . . . . . 6 ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) → (𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅)))
5251exlimiv 1928 . . . . 5 (∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) → (𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅)))
5331eqeq2d 2746 . . . . . . . 8 (𝑦 = ∅ → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨𝑥, ∅⟩))
54 eqeq1 2739 . . . . . . . . 9 (𝑦 = ∅ → (𝑦 = ∅ ↔ ∅ = ∅))
5554anbi1d 631 . . . . . . . 8 (𝑦 = ∅ → ((𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)) ↔ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))))
5653, 55anbi12d 632 . . . . . . 7 (𝑦 = ∅ → ((𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) ↔ (𝑋 = ⟨𝑥, ∅⟩ ∧ (∅ = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)))))
5743, 56spcev 3606 . . . . . 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 209 . . . 4 (∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) ↔ (𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅)))
6059exbii 1845 . . 3 (∃𝑥𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅)))
6129, 30, 603bitri 297 . 2 (𝑋 ∈ (𝑆‘∅) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅)))
6225satf0suc 35361 . . . . . . 7 (𝑧 ∈ ω → (𝑆‘suc 𝑧) = ((𝑆𝑧) ∪ {⟨𝑎, 𝑏⟩ ∣ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))}))
6362eleq2d 2825 . . . . . 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 4163 . . . . . . 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 5537 . . . . . . . 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 915 . . . . . 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 305 . . . . 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 480 . . . 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 484 . . . . . 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 2746 . . . . . . . . . . . . . . . 16 (𝑏 = ∅ → (𝑋 = ⟨𝑎, 𝑏⟩ ↔ 𝑋 = ⟨𝑎, ∅⟩))
7473biimpd 229 . . . . . . . . . . . . . . 15 (𝑏 = ∅ → (𝑋 = ⟨𝑎, 𝑏⟩ → 𝑋 = ⟨𝑎, ∅⟩))
7574adantr 480 . . . . . . . . . . . . . 14 ((𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢))) → (𝑋 = ⟨𝑎, 𝑏⟩ → 𝑋 = ⟨𝑎, ∅⟩))
7675impcom 407 . . . . . . . . . . . . 13 ((𝑋 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))) → 𝑋 = ⟨𝑎, ∅⟩)
77 eqidd 2736 . . . . . . . . . . . . . 14 ((𝑋 = ⟨𝑎, 𝑏⟩ ∧ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))) → ∅ = ∅)
78 simpr 484 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 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 511 . . . . . . . . . . . . 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 511 . . . . . . . . . . . 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 1928 . . . . . . . . . . 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 2739 . . . . . . . . . . . . . 14 (𝑏 = ∅ → (𝑏 = ∅ ↔ ∅ = ∅))
8483anbi1d 631 . . . . . . . . . . . . 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 632 . . . . . . . . . . . 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 3606 . . . . . . . . . . 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 209 . . . . . . . . . 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 1845 . . . . . . . . 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 2746 . . . . . . . . . 10 (𝑥 = 𝑎 → (𝑋 = ⟨𝑥, ∅⟩ ↔ 𝑋 = ⟨𝑎, ∅⟩))
92 eqeq1 2739 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ 𝑎 = ((1st𝑢)⊼𝑔(1st𝑣))))
9392rexbidv 3177 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ ∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣))))
94 eqeq1 2739 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (𝑥 = ∀𝑔𝑖(1st𝑢) ↔ 𝑎 = ∀𝑔𝑖(1st𝑢)))
9594rexbidv 3177 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢) ↔ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))
9693, 95orbi12d 918 . . . . . . . . . . . 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 3177 . . . . . . . . . . 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 630 . . . . . . . . . 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 632 . . . . . . . . 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 2034 . . . . . . . 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 289 . . . . . . 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 480 . . . . . 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 918 . . . . 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 1880 . . . . . 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 1009 . . . . . . . 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 224 . . . . . . 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 1845 . . . . . 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 277 . . . . 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 287 . . . 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 2825 . . . . . . . . 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 4163 . . . . . . . . . 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 2739 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → (𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
113112rexbidv 3177 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → (∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ ∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
114 eqeq1 2739 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → (𝑎 = ∀𝑔𝑖(1st𝑢) ↔ 𝑥 = ∀𝑔𝑖(1st𝑢)))
115114rexbidv 3177 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → (∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢) ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))
116113, 115orbi12d 918 . . . . . . . . . . . . . 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 3177 . . . . . . . . . . . . 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 639 . . . . . . . . . . . 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 5546 . . . . . . . . . . 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 912 . . . . . . . . . 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 275 . . . . . . . . 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 287 . . . . . . . 8 (𝑧 ∈ ω → (⟨𝑥, ∅⟩ ∈ (𝑆‘suc 𝑧) ↔ (⟨𝑥, ∅⟩ ∈ (𝑆𝑧) ∨ (∅ = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))))
123122anbi2d 630 . . . . . . 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 1919 . . . . . 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 223 . . . . 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 480 . . . 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 305 . . 3 ((𝑧 ∈ ω ∧ (𝑋 ∈ (𝑆𝑧) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧)))) → (𝑋 ∈ (𝑆‘suc 𝑧) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘suc 𝑧))))
128127ex 412 . 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 7919 1 (𝑁 ∈ ω → (𝑋 ∈ (𝑆𝑁) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑁))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wex 1776  wcel 2106  wrex 3068  cun 3961  c0 4339  cop 4637  {copab 5210  suc csuc 6388  cfv 6563  (class class class)co 7431  ωcom 7887  1st c1st 8011  𝑔cgoe 35318  𝑔cgna 35319  𝑔cgol 35320   Sat csat 35321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-map 8867  df-goel 35325  df-sat 35328
This theorem is referenced by:  fmlasuc  35371
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