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Theorem cnfinltrel 33557
Description: Less than for the Cantor normal form is a relation. (Contributed by ML, 24-Jun-2022.)
Hypotheses
Ref Expression
cnfin.1 𝐼 = {⟨∅, 1𝑜⟩}
cnfin.add + = (𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑛 ∈ (dom 𝑦 ∪ dom 𝑧) ↦ ((𝑦𝑛) +𝑜 (𝑧𝑛))))
cnfin.tr (𝜑 ↔ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐))
cnfin.ltadd (𝜓 ↔ (𝑥 ∈ (dom 𝑐 ∪ ran 𝑐) ∧ ∃𝑏 ∈ ran 𝑐 𝑦 = (𝑥 + 𝑏)))
cnfin.ltexp (𝜒 ↔ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩})))
cnfin.yrule 𝑌 = {⟨𝑥, 𝑦⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ 𝑐 ∨ (𝜑 ∨ (𝜓𝜒)))}
cnfin.lt < = ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)
cnfin.def 𝐶 = dom <
Assertion
Ref Expression
cnfinltrel Rel <
Distinct variable groups:   𝐼,𝑐   𝑎,𝑏,𝑐,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑛,𝑎,𝑏,𝑐)   𝜓(𝑥,𝑦,𝑧,𝑛,𝑎,𝑏,𝑐)   𝜒(𝑥,𝑦,𝑧,𝑛,𝑎,𝑏,𝑐)   𝐶(𝑥,𝑦,𝑧,𝑛,𝑎,𝑏,𝑐)   + (𝑥,𝑦,𝑧,𝑛,𝑎,𝑏,𝑐)   < (𝑥,𝑦,𝑧,𝑛,𝑎,𝑏,𝑐)   𝐼(𝑥,𝑦,𝑧,𝑛,𝑎,𝑏)   𝑌(𝑥,𝑦,𝑧,𝑛,𝑎,𝑏,𝑐)

Proof of Theorem cnfinltrel
Dummy variables 𝑘 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reluni 5443 . . 3 (Rel ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω) ↔ ∀𝑘 ∈ ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)Rel 𝑘)
2 frfnom 7766 . . . . 5 (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω) Fn ω
3 fvelrnb 6464 . . . . 5 ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω) Fn ω → (𝑘 ∈ ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω) ↔ ∃ ∈ ω ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) = 𝑘))
42, 3ax-mp 5 . . . 4 (𝑘 ∈ ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω) ↔ ∃ ∈ ω ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) = 𝑘)
5 0ex 4984 . . . . . . . . 9 ∅ ∈ V
6 cnfin.1 . . . . . . . . . 10 𝐼 = {⟨∅, 1𝑜⟩}
7 snex 5098 . . . . . . . . . 10 {⟨∅, 1𝑜⟩} ∈ V
86, 7eqeltri 2881 . . . . . . . . 9 𝐼 ∈ V
95, 8relsnop 5429 . . . . . . . 8 Rel {⟨∅, 𝐼⟩}
10 fveq2 6408 . . . . . . . . . . 11 ( = ∅ → ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) = ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘∅))
11 snex 5098 . . . . . . . . . . . 12 {⟨∅, 𝐼⟩} ∈ V
12 fr0g 7767 . . . . . . . . . . . 12 ({⟨∅, 𝐼⟩} ∈ V → ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘∅) = {⟨∅, 𝐼⟩})
1311, 12ax-mp 5 . . . . . . . . . . 11 ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘∅) = {⟨∅, 𝐼⟩}
1410, 13syl6eq 2856 . . . . . . . . . 10 ( = ∅ → ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) = {⟨∅, 𝐼⟩})
1514releqd 5405 . . . . . . . . 9 ( = ∅ → (Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) ↔ Rel {⟨∅, 𝐼⟩}))
165, 15sbcie 3668 . . . . . . . 8 ([∅ / ]Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) ↔ Rel {⟨∅, 𝐼⟩})
179, 16mpbir 222 . . . . . . 7 [∅ / ]Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘)
18 fvex 6421 . . . . . . . . . . . 12 ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) ∈ V
19 cnfin.yrule . . . . . . . . . . . . . 14 𝑌 = {⟨𝑥, 𝑦⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ 𝑐 ∨ (𝜑 ∨ (𝜓𝜒)))}
2019relopabi 5447 . . . . . . . . . . . . 13 Rel 𝑌
2120sbcth 3648 . . . . . . . . . . . 12 (((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) ∈ V → [((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐]Rel 𝑌)
2218, 21ax-mp 5 . . . . . . . . . . 11 [((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐]Rel 𝑌
23 sbcrel 5407 . . . . . . . . . . . 12 (((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) ∈ V → ([((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐]Rel 𝑌 ↔ Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐𝑌))
2418, 23ax-mp 5 . . . . . . . . . . 11 ([((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐]Rel 𝑌 ↔ Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐𝑌)
2522, 24mpbi 221 . . . . . . . . . 10 Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐𝑌
26 frsuc 7768 . . . . . . . . . . . 12 ( ∈ ω → ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘suc ) = ((𝑐 ∈ V ↦ 𝑌)‘((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘)))
27 cnfin.ltexp . . . . . . . . . . . . . . . . 17 (𝜒 ↔ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩})))
2827opabbii 4911 . . . . . . . . . . . . . . . 16 {⟨𝑥, 𝑦⟩ ∣ 𝜒} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))}
29 vex 3394 . . . . . . . . . . . . . . . . . . 19 𝑐 ∈ V
3029dmex 7329 . . . . . . . . . . . . . . . . . 18 dom 𝑐 ∈ V
3129rnex 7330 . . . . . . . . . . . . . . . . . 18 ran 𝑐 ∈ V
3230, 31ab2rexex 7389 . . . . . . . . . . . . . . . . 17 {𝑧 ∣ ∃𝑎 ∈ dom 𝑐𝑏 ∈ ran 𝑐 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩} ∈ V
33 df-opab 4907 . . . . . . . . . . . . . . . . . 18 {⟨𝑥, 𝑦⟩ ∣ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩})))}
34 19.42vv 2046 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑎𝑏(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))))
35342exbii 1934 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑥𝑦𝑎𝑏(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))))
36 vex 3394 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑎 ∈ V
37 vex 3394 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑏 ∈ V
3836, 37opeldm 5529 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (⟨𝑎, 𝑏⟩ ∈ 𝑐𝑎 ∈ dom 𝑐)
3936, 37opelrn 5558 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (⟨𝑎, 𝑏⟩ ∈ 𝑐𝑏 ∈ ran 𝑐)
4038, 39jca 503 . . . . . . . . . . . . . . . . . . . . . . . . 25 (⟨𝑎, 𝑏⟩ ∈ 𝑐 → (𝑎 ∈ dom 𝑐𝑏 ∈ ran 𝑐))
4140ad2antrl 710 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))) → (𝑎 ∈ dom 𝑐𝑏 ∈ ran 𝑐))
42 opeq12 4597 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}) → ⟨𝑥, 𝑦⟩ = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩)
4342eqeq2d 2816 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}) → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩))
4443biimpac 466 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩})) → 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩)
4544adantrl 698 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))) → 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩)
4641, 45jca 503 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))) → ((𝑎 ∈ dom 𝑐𝑏 ∈ ran 𝑐) ∧ 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩))
47462eximi 1920 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑎𝑏(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))) → ∃𝑎𝑏((𝑎 ∈ dom 𝑐𝑏 ∈ ran 𝑐) ∧ 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩))
4847exlimivv 2023 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑥𝑦𝑎𝑏(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))) → ∃𝑎𝑏((𝑎 ∈ dom 𝑐𝑏 ∈ ran 𝑐) ∧ 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩))
4935, 48sylbir 226 . . . . . . . . . . . . . . . . . . . 20 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))) → ∃𝑎𝑏((𝑎 ∈ dom 𝑐𝑏 ∈ ran 𝑐) ∧ 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩))
50 r2ex 3249 . . . . . . . . . . . . . . . . . . . 20 (∃𝑎 ∈ dom 𝑐𝑏 ∈ ran 𝑐 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩ ↔ ∃𝑎𝑏((𝑎 ∈ dom 𝑐𝑏 ∈ ran 𝑐) ∧ 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩))
5149, 50sylibr 225 . . . . . . . . . . . . . . . . . . 19 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))) → ∃𝑎 ∈ dom 𝑐𝑏 ∈ ran 𝑐 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩)
5251ss2abi 3871 . . . . . . . . . . . . . . . . . 18 {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩})))} ⊆ {𝑧 ∣ ∃𝑎 ∈ dom 𝑐𝑏 ∈ ran 𝑐 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩}
5333, 52eqsstri 3832 . . . . . . . . . . . . . . . . 17 {⟨𝑥, 𝑦⟩ ∣ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))} ⊆ {𝑧 ∣ ∃𝑎 ∈ dom 𝑐𝑏 ∈ ran 𝑐 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩}
5432, 53ssexi 4998 . . . . . . . . . . . . . . . 16 {⟨𝑥, 𝑦⟩ ∣ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))} ∈ V
5528, 54eqeltri 2881 . . . . . . . . . . . . . . 15 {⟨𝑥, 𝑦⟩ ∣ 𝜒} ∈ V
56 unopab 4922 . . . . . . . . . . . . . . . . 17 ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜒}) = {⟨𝑥, 𝑦⟩ ∣ (𝜓𝜒)}
57 cnfin.ltadd . . . . . . . . . . . . . . . . . . . 20 (𝜓 ↔ (𝑥 ∈ (dom 𝑐 ∪ ran 𝑐) ∧ ∃𝑏 ∈ ran 𝑐 𝑦 = (𝑥 + 𝑏)))
5857opabbii 4911 . . . . . . . . . . . . . . . . . . 19 {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (dom 𝑐 ∪ ran 𝑐) ∧ ∃𝑏 ∈ ran 𝑐 𝑦 = (𝑥 + 𝑏))}
5930, 31unex 7186 . . . . . . . . . . . . . . . . . . . 20 (dom 𝑐 ∪ ran 𝑐) ∈ V
60 nfcv 2948 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑦ran 𝑐
61 nfcv 2948 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑦𝑥
62 cnfin.add . . . . . . . . . . . . . . . . . . . . . . . . . . 27 + = (𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑛 ∈ (dom 𝑦 ∪ dom 𝑧) ↦ ((𝑦𝑛) +𝑜 (𝑧𝑛))))
63 nfmpt21 6952 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑦(𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑛 ∈ (dom 𝑦 ∪ dom 𝑧) ↦ ((𝑦𝑛) +𝑜 (𝑧𝑛))))
6462, 63nfcxfr 2946 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑦 +
65 nfcv 2948 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑦𝑏
6661, 64, 65nfov 6904 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑦(𝑥 + 𝑏)
6766nfeq2 2964 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑦 𝑘 = (𝑥 + 𝑏)
6860, 67nfrex 3194 . . . . . . . . . . . . . . . . . . . . . . 23 𝑦𝑏 ∈ ran 𝑐 𝑘 = (𝑥 + 𝑏)
69 nfv 2005 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘𝑏 ∈ ran 𝑐 𝑦 = (𝑥 + 𝑏)
70 eqeq1 2810 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 𝑦 → (𝑘 = (𝑥 + 𝑏) ↔ 𝑦 = (𝑥 + 𝑏)))
7170rexbidv 3240 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝑦 → (∃𝑏 ∈ ran 𝑐 𝑘 = (𝑥 + 𝑏) ↔ ∃𝑏 ∈ ran 𝑐 𝑦 = (𝑥 + 𝑏)))
7268, 69, 71cbvab 2930 . . . . . . . . . . . . . . . . . . . . . 22 {𝑘 ∣ ∃𝑏 ∈ ran 𝑐 𝑘 = (𝑥 + 𝑏)} = {𝑦 ∣ ∃𝑏 ∈ ran 𝑐 𝑦 = (𝑥 + 𝑏)}
73 abrexexg 7370 . . . . . . . . . . . . . . . . . . . . . . 23 (ran 𝑐 ∈ V → {𝑘 ∣ ∃𝑏 ∈ ran 𝑐 𝑘 = (𝑥 + 𝑏)} ∈ V)
7431, 73ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 {𝑘 ∣ ∃𝑏 ∈ ran 𝑐 𝑘 = (𝑥 + 𝑏)} ∈ V
7572, 74eqeltrri 2882 . . . . . . . . . . . . . . . . . . . . 21 {𝑦 ∣ ∃𝑏 ∈ ran 𝑐 𝑦 = (𝑥 + 𝑏)} ∈ V
7675a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (dom 𝑐 ∪ ran 𝑐) → {𝑦 ∣ ∃𝑏 ∈ ran 𝑐 𝑦 = (𝑥 + 𝑏)} ∈ V)
7759, 76opabex3 7376 . . . . . . . . . . . . . . . . . . 19 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (dom 𝑐 ∪ ran 𝑐) ∧ ∃𝑏 ∈ ran 𝑐 𝑦 = (𝑥 + 𝑏))} ∈ V
7858, 77eqeltri 2881 . . . . . . . . . . . . . . . . . 18 {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ V
79 unexg 7189 . . . . . . . . . . . . . . . . . 18 (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ V ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜒} ∈ V) → ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜒}) ∈ V)
8078, 79mpan 673 . . . . . . . . . . . . . . . . 17 ({⟨𝑥, 𝑦⟩ ∣ 𝜒} ∈ V → ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜒}) ∈ V)
8156, 80syl5eqelr 2890 . . . . . . . . . . . . . . . 16 ({⟨𝑥, 𝑦⟩ ∣ 𝜒} ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝜓𝜒)} ∈ V)
82 unopab 4922 . . . . . . . . . . . . . . . . 17 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝜓𝜒)}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ (𝜓𝜒))}
83 cnfin.tr . . . . . . . . . . . . . . . . . . . 20 (𝜑 ↔ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐))
8483opabbii 4911 . . . . . . . . . . . . . . . . . . 19 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐)}
8530, 31xpex 7192 . . . . . . . . . . . . . . . . . . . 20 (dom 𝑐 × ran 𝑐) ∈ V
86 df-opab 4907 . . . . . . . . . . . . . . . . . . . . 21 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐)} = {𝑙 ∣ ∃𝑥𝑦(𝑙 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐))}
87 eleq1 2873 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑙 = ⟨𝑥, 𝑦⟩ → (𝑙 ∈ (dom 𝑐 × ran 𝑐) ↔ ⟨𝑥, 𝑦⟩ ∈ (dom 𝑐 × ran 𝑐)))
8887biimprd 239 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑙 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ (dom 𝑐 × ran 𝑐) → 𝑙 ∈ (dom 𝑐 × ran 𝑐)))
89 vex 3394 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑥 ∈ V
90 vex 3394 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑧 ∈ V
9189, 90opeldm 5529 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (⟨𝑥, 𝑧⟩ ∈ 𝑐𝑥 ∈ dom 𝑐)
92 vex 3394 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑦 ∈ V
9390, 92opelrn 5558 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (⟨𝑧, 𝑦⟩ ∈ 𝑐𝑦 ∈ ran 𝑐)
94 opelxpi 5348 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ dom 𝑐𝑦 ∈ ran 𝑐) → ⟨𝑥, 𝑦⟩ ∈ (dom 𝑐 × ran 𝑐))
9591, 93, 94syl2an 585 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐) → ⟨𝑥, 𝑦⟩ ∈ (dom 𝑐 × ran 𝑐))
9695exlimiv 2021 . . . . . . . . . . . . . . . . . . . . . . . 24 (∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐) → ⟨𝑥, 𝑦⟩ ∈ (dom 𝑐 × ran 𝑐))
9788, 96impel 497 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑙 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐)) → 𝑙 ∈ (dom 𝑐 × ran 𝑐))
9897exlimivv 2023 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑥𝑦(𝑙 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐)) → 𝑙 ∈ (dom 𝑐 × ran 𝑐))
9998abssi 3874 . . . . . . . . . . . . . . . . . . . . 21 {𝑙 ∣ ∃𝑥𝑦(𝑙 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐))} ⊆ (dom 𝑐 × ran 𝑐)
10086, 99eqsstri 3832 . . . . . . . . . . . . . . . . . . . 20 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐)} ⊆ (dom 𝑐 × ran 𝑐)
10185, 100ssexi 4998 . . . . . . . . . . . . . . . . . . 19 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐)} ∈ V
10284, 101eqeltri 2881 . . . . . . . . . . . . . . . . . 18 {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V
103 unexg 7189 . . . . . . . . . . . . . . . . . 18 (({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V ∧ {⟨𝑥, 𝑦⟩ ∣ (𝜓𝜒)} ∈ V) → ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝜓𝜒)}) ∈ V)
104102, 103mpan 673 . . . . . . . . . . . . . . . . 17 ({⟨𝑥, 𝑦⟩ ∣ (𝜓𝜒)} ∈ V → ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝜓𝜒)}) ∈ V)
10582, 104syl5eqelr 2890 . . . . . . . . . . . . . . . 16 ({⟨𝑥, 𝑦⟩ ∣ (𝜓𝜒)} ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ (𝜓𝜒))} ∈ V)
106 unopab 4922 . . . . . . . . . . . . . . . . . 18 ({⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝑐} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ (𝜓𝜒))}) = {⟨𝑥, 𝑦⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ 𝑐 ∨ (𝜑 ∨ (𝜓𝜒)))}
107106, 19eqtr4i 2831 . . . . . . . . . . . . . . . . 17 ({⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝑐} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ (𝜓𝜒))}) = 𝑌
108 df-opab 4907 . . . . . . . . . . . . . . . . . . . 20 {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝑐} = {𝑙 ∣ ∃𝑥𝑦(𝑙 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑐)}
109 eleq1 2873 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑙 = ⟨𝑥, 𝑦⟩ → (𝑙𝑐 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑐))
110109biimprd 239 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ 𝑐𝑙𝑐))
111 id 22 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨𝑥, 𝑦⟩ ∈ 𝑐 → ⟨𝑥, 𝑦⟩ ∈ 𝑐)
112110, 111impel 497 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑙 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑐) → 𝑙𝑐)
113112exlimivv 2023 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑥𝑦(𝑙 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑐) → 𝑙𝑐)
114113abssi 3874 . . . . . . . . . . . . . . . . . . . 20 {𝑙 ∣ ∃𝑥𝑦(𝑙 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑐)} ⊆ 𝑐
115108, 114eqsstri 3832 . . . . . . . . . . . . . . . . . . 19 {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝑐} ⊆ 𝑐
11629, 115ssexi 4998 . . . . . . . . . . . . . . . . . 18 {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝑐} ∈ V
117 unexg 7189 . . . . . . . . . . . . . . . . . 18 (({⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝑐} ∈ V ∧ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ (𝜓𝜒))} ∈ V) → ({⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝑐} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ (𝜓𝜒))}) ∈ V)
118116, 117mpan 673 . . . . . . . . . . . . . . . . 17 ({⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ (𝜓𝜒))} ∈ V → ({⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝑐} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ (𝜓𝜒))}) ∈ V)
119107, 118syl5eqelr 2890 . . . . . . . . . . . . . . . 16 ({⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ (𝜓𝜒))} ∈ V → 𝑌 ∈ V)
12081, 105, 1193syl 18 . . . . . . . . . . . . . . 15 ({⟨𝑥, 𝑦⟩ ∣ 𝜒} ∈ V → 𝑌 ∈ V)
12155, 120ax-mp 5 . . . . . . . . . . . . . 14 𝑌 ∈ V
122121csbex 4988 . . . . . . . . . . . . 13 ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐𝑌 ∈ V
123 eqid 2806 . . . . . . . . . . . . . 14 (𝑐 ∈ V ↦ 𝑌) = (𝑐 ∈ V ↦ 𝑌)
124123fvmpts 6506 . . . . . . . . . . . . 13 ((((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) ∈ V ∧ ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐𝑌 ∈ V) → ((𝑐 ∈ V ↦ 𝑌)‘((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘)) = ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐𝑌)
12518, 122, 124mp2an 675 . . . . . . . . . . . 12 ((𝑐 ∈ V ↦ 𝑌)‘((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘)) = ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐𝑌
12626, 125syl6eq 2856 . . . . . . . . . . 11 ( ∈ ω → ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘suc ) = ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐𝑌)
127126releqd 5405 . . . . . . . . . 10 ( ∈ ω → (Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘suc ) ↔ Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐𝑌))
12825, 127mpbiri 249 . . . . . . . . 9 ( ∈ ω → Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘suc ))
129 vex 3394 . . . . . . . . . . . 12 ∈ V
130129sucex 7241 . . . . . . . . . . 11 suc ∈ V
131 sbcrel 5407 . . . . . . . . . . 11 (suc ∈ V → ([suc / ]Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) ↔ Rel suc / ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘)))
132130, 131ax-mp 5 . . . . . . . . . 10 ([suc / ]Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) ↔ Rel suc / ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘))
133 csbfv 6453 . . . . . . . . . . 11 suc / ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) = ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘suc )
134133releqi 5404 . . . . . . . . . 10 (Rel suc / ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) ↔ Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘suc ))
135132, 134bitri 266 . . . . . . . . 9 ([suc / ]Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) ↔ Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘suc ))
136128, 135sylibr 225 . . . . . . . 8 ( ∈ ω → [suc / ]Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘))
137136a1d 25 . . . . . . 7 ( ∈ ω → (Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) → [suc / ]Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘)))
13817, 137findes 7326 . . . . . 6 ( ∈ ω → Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘))
139 releq 5403 . . . . . 6 (((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) = 𝑘 → (Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) ↔ Rel 𝑘))
140138, 139syl5ibcom 236 . . . . 5 ( ∈ ω → (((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) = 𝑘 → Rel 𝑘))
141140rexlimiv 3215 . . . 4 (∃ ∈ ω ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) = 𝑘 → Rel 𝑘)
1424, 141sylbi 208 . . 3 (𝑘 ∈ ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω) → Rel 𝑘)
1431, 142mprgbir 3115 . 2 Rel ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)
144 cnfin.lt . . 3 < = ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)
145144releqi 5404 . 2 (Rel < ↔ Rel ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω))
146143, 145mpbir 222 1 Rel <
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  wo 865   = wceq 1637  wex 1859  wcel 2156  {cab 2792  wrex 3097  Vcvv 3391  [wsbc 3633  csb 3728  cun 3767  c0 4116  {csn 4370  cop 4376   cuni 4630  {copab 4906  cmpt 4923   × cxp 5309  dom cdm 5311  ran crn 5312  cres 5313  Rel wrel 5316  suc csuc 5938   Fn wfn 6096  cfv 6101  (class class class)co 6874  cmpt2 6876  ωcom 7295  reccrdg 7741  1𝑜c1o 7789   +𝑜 coa 7793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-om 7296  df-wrecs 7642  df-recs 7704  df-rdg 7742
This theorem is referenced by: (None)
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