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Theorem satf0op 35382
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 6906 . . . 4 (𝑦 = ∅ → (𝑆𝑦) = (𝑆‘∅))
21eleq2d 2827 . . 3 (𝑦 = ∅ → (𝑋 ∈ (𝑆𝑦) ↔ 𝑋 ∈ (𝑆‘∅)))
31eleq2d 2827 . . . . 5 (𝑦 = ∅ → (⟨𝑥, ∅⟩ ∈ (𝑆𝑦) ↔ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅)))
43anbi2d 630 . . . 4 (𝑦 = ∅ → ((𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦)) ↔ (𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅))))
54exbidv 1921 . . 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 6906 . . . 4 (𝑦 = 𝑧 → (𝑆𝑦) = (𝑆𝑧))
87eleq2d 2827 . . 3 (𝑦 = 𝑧 → (𝑋 ∈ (𝑆𝑦) ↔ 𝑋 ∈ (𝑆𝑧)))
97eleq2d 2827 . . . . 5 (𝑦 = 𝑧 → (⟨𝑥, ∅⟩ ∈ (𝑆𝑦) ↔ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧)))
109anbi2d 630 . . . 4 (𝑦 = 𝑧 → ((𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦)) ↔ (𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑧))))
1110exbidv 1921 . . 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 6906 . . . 4 (𝑦 = suc 𝑧 → (𝑆𝑦) = (𝑆‘suc 𝑧))
1413eleq2d 2827 . . 3 (𝑦 = suc 𝑧 → (𝑋 ∈ (𝑆𝑦) ↔ 𝑋 ∈ (𝑆‘suc 𝑧)))
1513eleq2d 2827 . . . . 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 1921 . . 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 6906 . . . 4 (𝑦 = 𝑁 → (𝑆𝑦) = (𝑆𝑁))
2019eleq2d 2827 . . 3 (𝑦 = 𝑁 → (𝑋 ∈ (𝑆𝑦) ↔ 𝑋 ∈ (𝑆𝑁)))
2119eleq2d 2827 . . . . 5 (𝑦 = 𝑁 → (⟨𝑥, ∅⟩ ∈ (𝑆𝑦) ↔ ⟨𝑥, ∅⟩ ∈ (𝑆𝑁)))
2221anbi2d 630 . . . 4 (𝑦 = 𝑁 → ((𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑦)) ↔ (𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑁))))
2322exbidv 1921 . . 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 6907 . . . . 5 (𝑆‘∅) = ((∅ Sat ∅)‘∅)
27 satf00 35379 . . . . 5 ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}
2826, 27eqtri 2765 . . . 4 (𝑆‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}
2928eleq2i 2833 . . 3 (𝑋 ∈ (𝑆‘∅) ↔ 𝑋 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})
30 elopab 5532 . . 3 (𝑋 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))} ↔ ∃𝑥𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))))
31 opeq2 4874 . . . . . . . . . . 11 (𝑦 = ∅ → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ∅⟩)
3231adantr 480 . . . . . . . . . 10 ((𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)) → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ∅⟩)
3332eqeq2d 2748 . . . . . . . . 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 2738 . . . . . . . . . 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 35379 . . . . . . . . . . 11 ((∅ Sat ∅)‘∅) = {⟨𝑦, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗))}
4026, 39eqtri 2765 . . . . . . . . . 10 (𝑆‘∅) = {⟨𝑦, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗))}
4140eleq2i 2833 . . . . . . . . 9 (⟨𝑥, ∅⟩ ∈ (𝑆‘∅) ↔ ⟨𝑥, ∅⟩ ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑧 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗))})
42 vex 3484 . . . . . . . . . 10 𝑥 ∈ V
43 0ex 5307 . . . . . . . . . 10 ∅ ∈ V
44 eqeq1 2741 . . . . . . . . . . 11 (𝑧 = ∅ → (𝑧 = ∅ ↔ ∅ = ∅))
45 eqeq1 2741 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (𝑦 = (𝑖𝑔𝑗) ↔ 𝑥 = (𝑖𝑔𝑗)))
46452rexbidv 3222 . . . . . . . . . . 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 5541 . . . . . . . . 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 1930 . . . . 5 (∃𝑦(𝑋 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))) → (𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘∅)))
5331eqeq2d 2748 . . . . . . . 8 (𝑦 = ∅ → (𝑋 = ⟨𝑥, 𝑦⟩ ↔ 𝑋 = ⟨𝑥, ∅⟩))
54 eqeq1 2741 . . . . . . . . 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 1848 . . 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 35381 . . . . . . 7 (𝑧 ∈ ω → (𝑆‘suc 𝑧) = ((𝑆𝑧) ∪ {⟨𝑎, 𝑏⟩ ∣ (𝑏 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑧)(∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))}))
6362eleq2d 2827 . . . . . 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 4153 . . . . . . 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 5532 . . . . . . . 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 916 . . . . . 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 4874 . . . . . . . . . . . . . . . . 17 (𝑏 = ∅ → ⟨𝑎, 𝑏⟩ = ⟨𝑎, ∅⟩)
7372eqeq2d 2748 . . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . 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 1930 . . . . . . . . . . 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 2741 . . . . . . . . . . . . . 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 1848 . . . . . . . . 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 4873 . . . . . . . . . . 11 (𝑥 = 𝑎 → ⟨𝑥, ∅⟩ = ⟨𝑎, ∅⟩)
9190eqeq2d 2748 . . . . . . . . . 10 (𝑥 = 𝑎 → (𝑋 = ⟨𝑥, ∅⟩ ↔ 𝑋 = ⟨𝑎, ∅⟩))
92 eqeq1 2741 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ 𝑎 = ((1st𝑢)⊼𝑔(1st𝑣))))
9392rexbidv 3179 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ ∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣))))
94 eqeq1 2741 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (𝑥 = ∀𝑔𝑖(1st𝑢) ↔ 𝑎 = ∀𝑔𝑖(1st𝑢)))
9594rexbidv 3179 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢) ↔ ∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢)))
9693, 95orbi12d 919 . . . . . . . . . . . 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 3179 . . . . . . . . . . 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 2036 . . . . . . . 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 919 . . . . 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 1882 . . . . . 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 1010 . . . . . . . 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 1848 . . . . . 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 2827 . . . . . . . . 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 4153 . . . . . . . . . 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 2741 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → (𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
113112rexbidv 3179 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → (∃𝑣 ∈ (𝑆𝑧)𝑎 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ ∃𝑣 ∈ (𝑆𝑧)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
114 eqeq1 2741 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → (𝑎 = ∀𝑔𝑖(1st𝑢) ↔ 𝑥 = ∀𝑔𝑖(1st𝑢)))
115114rexbidv 3179 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → (∃𝑖 ∈ ω 𝑎 = ∀𝑔𝑖(1st𝑢) ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))
116113, 115orbi12d 919 . . . . . . . . . . . . . 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 3179 . . . . . . . . . . . . 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 5541 . . . . . . . . . . 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 913 . . . . . . . . . 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 1921 . . . . . 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 7918 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 848   = wceq 1540  wex 1779  wcel 2108  wrex 3070  cun 3949  c0 4333  cop 4632  {copab 5205  suc csuc 6386  cfv 6561  (class class class)co 7431  ωcom 7887  1st c1st 8012  𝑔cgoe 35338  𝑔cgna 35339  𝑔cgol 35340   Sat csat 35341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-map 8868  df-goel 35345  df-sat 35348
This theorem is referenced by:  fmlasuc  35391
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