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Theorem satfvsuclem2 35345
Description: Lemma 2 for satfvsuc 35346. (Contributed by AV, 8-Oct-2023.)
Hypothesis
Ref Expression
satfv0.s 𝑆 = (𝑀 Sat 𝐸)
Assertion
Ref Expression
satfvsuclem2 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))} ∈ V)
Distinct variable groups:   𝐸,𝑎,𝑖,𝑢,𝑣,𝑥,𝑦,𝑧   𝑀,𝑎,𝑖,𝑢,𝑣,𝑥,𝑦,𝑧   𝑢,𝑁,𝑣,𝑥,𝑦   𝑢,𝑆,𝑣,𝑥   𝑢,𝑉,𝑦   𝑢,𝑊,𝑦
Allowed substitution hints:   𝑆(𝑦,𝑧,𝑖,𝑎)   𝑁(𝑧,𝑖,𝑎)   𝑉(𝑥,𝑧,𝑣,𝑖,𝑎)   𝑊(𝑥,𝑧,𝑣,𝑖,𝑎)

Proof of Theorem satfvsuclem2
StepHypRef Expression
1 r19.41v 3187 . . . . . . 7 (∃𝑣 ∈ (𝑆𝑁)((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∧ 𝑦 ∈ 𝒫 (𝑀m ω)) ↔ (∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∧ 𝑦 ∈ 𝒫 (𝑀m ω)))
2 r19.41v 3187 . . . . . . 7 (∃𝑖 ∈ ω ((𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ∧ 𝑦 ∈ 𝒫 (𝑀m ω)) ↔ (∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ∧ 𝑦 ∈ 𝒫 (𝑀m ω)))
31, 2orbi12i 914 . . . . . 6 ((∃𝑣 ∈ (𝑆𝑁)((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∧ 𝑦 ∈ 𝒫 (𝑀m ω)) ∨ ∃𝑖 ∈ ω ((𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ∧ 𝑦 ∈ 𝒫 (𝑀m ω))) ↔ ((∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∧ 𝑦 ∈ 𝒫 (𝑀m ω)) ∨ (∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ∧ 𝑦 ∈ 𝒫 (𝑀m ω))))
4 ovex 7464 . . . . . . . . . . . 12 (𝑀m ω) ∈ V
5 difelpw 5360 . . . . . . . . . . . 12 ((𝑀m ω) ∈ V → ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ∈ 𝒫 (𝑀m ω))
64, 5ax-mp 5 . . . . . . . . . . 11 ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ∈ 𝒫 (𝑀m ω)
7 eleq1 2827 . . . . . . . . . . 11 (𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) → (𝑦 ∈ 𝒫 (𝑀m ω) ↔ ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ∈ 𝒫 (𝑀m ω)))
86, 7mpbiri 258 . . . . . . . . . 10 (𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) → 𝑦 ∈ 𝒫 (𝑀m ω))
98pm4.71i 559 . . . . . . . . 9 (𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ↔ (𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ∧ 𝑦 ∈ 𝒫 (𝑀m ω)))
109bianass 642 . . . . . . . 8 ((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ↔ ((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∧ 𝑦 ∈ 𝒫 (𝑀m ω)))
1110rexbii 3092 . . . . . . 7 (∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ↔ ∃𝑣 ∈ (𝑆𝑁)((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∧ 𝑦 ∈ 𝒫 (𝑀m ω)))
12 rabelpw 5342 . . . . . . . . . . . 12 ((𝑀m ω) ∈ V → {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} ∈ 𝒫 (𝑀m ω))
134, 12ax-mp 5 . . . . . . . . . . 11 {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} ∈ 𝒫 (𝑀m ω)
14 eleq1 2827 . . . . . . . . . . 11 (𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} → (𝑦 ∈ 𝒫 (𝑀m ω) ↔ {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} ∈ 𝒫 (𝑀m ω)))
1513, 14mpbiri 258 . . . . . . . . . 10 (𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} → 𝑦 ∈ 𝒫 (𝑀m ω))
1615pm4.71i 559 . . . . . . . . 9 (𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} ↔ (𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} ∧ 𝑦 ∈ 𝒫 (𝑀m ω)))
1716bianass 642 . . . . . . . 8 ((𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ↔ ((𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ∧ 𝑦 ∈ 𝒫 (𝑀m ω)))
1817rexbii 3092 . . . . . . 7 (∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ↔ ∃𝑖 ∈ ω ((𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ∧ 𝑦 ∈ 𝒫 (𝑀m ω)))
1911, 18orbi12i 914 . . . . . 6 ((∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ↔ (∃𝑣 ∈ (𝑆𝑁)((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∧ 𝑦 ∈ 𝒫 (𝑀m ω)) ∨ ∃𝑖 ∈ ω ((𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ∧ 𝑦 ∈ 𝒫 (𝑀m ω))))
20 andir 1010 . . . . . 6 (((∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∧ 𝑦 ∈ 𝒫 (𝑀m ω)) ↔ ((∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∧ 𝑦 ∈ 𝒫 (𝑀m ω)) ∨ (∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ∧ 𝑦 ∈ 𝒫 (𝑀m ω))))
213, 19, 203bitr4i 303 . . . . 5 ((∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ↔ ((∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∧ 𝑦 ∈ 𝒫 (𝑀m ω)))
2221rexbii 3092 . . . 4 (∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ↔ ∃𝑢 ∈ (𝑆𝑁)((∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∧ 𝑦 ∈ 𝒫 (𝑀m ω)))
23 r19.41v 3187 . . . 4 (∃𝑢 ∈ (𝑆𝑁)((∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∧ 𝑦 ∈ 𝒫 (𝑀m ω)) ↔ (∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∧ 𝑦 ∈ 𝒫 (𝑀m ω)))
2422, 23bitri 275 . . 3 (∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ↔ (∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∧ 𝑦 ∈ 𝒫 (𝑀m ω)))
2524opabbii 5215 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))} = {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∧ 𝑦 ∈ 𝒫 (𝑀m ω))}
26 satfv0.s . . 3 𝑆 = (𝑀 Sat 𝐸)
2726satfvsuclem1 35344 . 2 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∧ 𝑦 ∈ 𝒫 (𝑀m ω))} ∈ V)
2825, 27eqeltrid 2843 1 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  cdif 3960  cun 3961  cin 3962  𝒫 cpw 4605  {csn 4631  cop 4637  {copab 5210  cres 5691  cfv 6563  (class class class)co 7431  ωcom 7887  1st c1st 8011  2nd c2nd 8012  m cmap 8865  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-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  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-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fv 6571  df-ov 7434  df-om 7888
This theorem is referenced by:  satfvsuc  35346
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