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Theorem bnj964 32203
 Description: Technical lemma for bnj69 32270. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj964.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj964.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj964.5 (𝜓′[𝑝 / 𝑛]𝜓)
bnj964.8 (𝜓″[𝐺 / 𝑓]𝜓′)
bnj964.12 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
bnj964.13 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
bnj964.96 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
bnj964.165 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
Assertion
Ref Expression
bnj964 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜓″)
Distinct variable groups:   𝐴,𝑓,𝑖,𝑛   𝐷,𝑖   𝑖,𝐺   𝑅,𝑓,𝑖,𝑛   𝑖,𝑋   𝑓,𝑝,𝑖   𝑦,𝑓,𝑖,𝑛   𝑖,𝑚   𝜑,𝑖
Allowed substitution hints:   𝜑(𝑦,𝑓,𝑚,𝑛,𝑝)   𝜓(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑦,𝑚,𝑝)   𝐶(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑓,𝑚,𝑛,𝑝)   𝑅(𝑦,𝑚,𝑝)   𝐺(𝑦,𝑓,𝑚,𝑛,𝑝)   𝑋(𝑦,𝑓,𝑚,𝑛,𝑝)   𝜓′(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓″(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj964
StepHypRef Expression
1 nfv 1908 . . . 4 𝑖(𝑅 FrSe 𝐴𝑋𝐴)
2 bnj964.2 . . . . . . . 8 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
32bnj1095 32041 . . . . . . 7 (𝜓 → ∀𝑖𝜓)
4 bnj964.3 . . . . . . 7 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
53, 4bnj1096 32042 . . . . . 6 (𝜒 → ∀𝑖𝜒)
65nf5i 2143 . . . . 5 𝑖𝜒
7 nfv 1908 . . . . 5 𝑖 𝑛 = suc 𝑚
8 nfv 1908 . . . . 5 𝑖 𝑝 = suc 𝑛
96, 7, 8nf3an 1895 . . . 4 𝑖(𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)
101, 9nfan 1893 . . 3 𝑖((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛))
11 bnj255 31963 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝)))
12 bnj645 32009 . . . . . . 7 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝) → suc 𝑖𝑝)
13 simp3 1132 . . . . . . . 8 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝑝 = suc 𝑛)
1413bnj706 32013 . . . . . . 7 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝) → 𝑝 = suc 𝑛)
15 eleq2 2899 . . . . . . . . 9 (𝑝 = suc 𝑛 → (suc 𝑖𝑝 ↔ suc 𝑖 ∈ suc 𝑛))
1615biimpac 481 . . . . . . . 8 ((suc 𝑖𝑝𝑝 = suc 𝑛) → suc 𝑖 ∈ suc 𝑛)
17 elsuci 6250 . . . . . . . . 9 (suc 𝑖 ∈ suc 𝑛 → (suc 𝑖𝑛 ∨ suc 𝑖 = 𝑛))
18 eqcom 2826 . . . . . . . . . 10 (suc 𝑖 = 𝑛𝑛 = suc 𝑖)
1918orbi2i 908 . . . . . . . . 9 ((suc 𝑖𝑛 ∨ suc 𝑖 = 𝑛) ↔ (suc 𝑖𝑛𝑛 = suc 𝑖))
2017, 19sylib 220 . . . . . . . 8 (suc 𝑖 ∈ suc 𝑛 → (suc 𝑖𝑛𝑛 = suc 𝑖))
2116, 20syl 17 . . . . . . 7 ((suc 𝑖𝑝𝑝 = suc 𝑛) → (suc 𝑖𝑛𝑛 = suc 𝑖))
2212, 14, 21syl2anc 586 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝) → (suc 𝑖𝑛𝑛 = suc 𝑖))
23 df-3an 1083 . . . . . . . . . . . . 13 ((𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛) ↔ ((𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ suc 𝑖𝑛))
24233anbi3i 1153 . . . . . . . . . . . 12 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ ((𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ suc 𝑖𝑛)))
25 bnj255 31963 . . . . . . . . . . . 12 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ suc 𝑖𝑛) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ ((𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ suc 𝑖𝑛)))
2624, 25bitr4i 280 . . . . . . . . . . 11 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ suc 𝑖𝑛))
27 bnj345 31972 . . . . . . . . . . 11 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ suc 𝑖𝑛) ↔ (suc 𝑖𝑛 ∧ (𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝)))
28 bnj252 31961 . . . . . . . . . . 11 ((suc 𝑖𝑛 ∧ (𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝)) ↔ (suc 𝑖𝑛 ∧ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝))))
2926, 27, 283bitri 299 . . . . . . . . . 10 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) ↔ (suc 𝑖𝑛 ∧ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝))))
3011anbi2i 624 . . . . . . . . . 10 ((suc 𝑖𝑛 ∧ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝)) ↔ (suc 𝑖𝑛 ∧ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝))))
3129, 30bitr4i 280 . . . . . . . . 9 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) ↔ (suc 𝑖𝑛 ∧ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝)))
32 bnj964.96 . . . . . . . . 9 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
3331, 32sylbir 237 . . . . . . . 8 ((suc 𝑖𝑛 ∧ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
3433ex 415 . . . . . . 7 (suc 𝑖𝑛 → (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
35 df-3an 1083 . . . . . . . . . . . . 13 ((𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖) ↔ ((𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ 𝑛 = suc 𝑖))
36353anbi3i 1153 . . . . . . . . . . . 12 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ ((𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ 𝑛 = suc 𝑖)))
37 bnj255 31963 . . . . . . . . . . . 12 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ 𝑛 = suc 𝑖) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ ((𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ 𝑛 = suc 𝑖)))
3836, 37bitr4i 280 . . . . . . . . . . 11 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ 𝑛 = suc 𝑖))
39 bnj345 31972 . . . . . . . . . . 11 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ 𝑛 = suc 𝑖) ↔ (𝑛 = suc 𝑖 ∧ (𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝)))
40 bnj252 31961 . . . . . . . . . . 11 ((𝑛 = suc 𝑖 ∧ (𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝)) ↔ (𝑛 = suc 𝑖 ∧ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝))))
4138, 39, 403bitri 299 . . . . . . . . . 10 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) ↔ (𝑛 = suc 𝑖 ∧ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝))))
4211anbi2i 624 . . . . . . . . . 10 ((𝑛 = suc 𝑖 ∧ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝)) ↔ (𝑛 = suc 𝑖 ∧ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝))))
4341, 42bitr4i 280 . . . . . . . . 9 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) ↔ (𝑛 = suc 𝑖 ∧ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝)))
44 bnj964.165 . . . . . . . . 9 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
4543, 44sylbir 237 . . . . . . . 8 ((𝑛 = suc 𝑖 ∧ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
4645ex 415 . . . . . . 7 (𝑛 = suc 𝑖 → (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
4734, 46jaoi 853 . . . . . 6 ((suc 𝑖𝑛𝑛 = suc 𝑖) → (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
4822, 47mpcom 38 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
4911, 48sylbir 237 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
50493expia 1115 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → ((𝑖 ∈ ω ∧ suc 𝑖𝑝) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
5110, 50alrimi 2205 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → ∀𝑖((𝑖 ∈ ω ∧ suc 𝑖𝑝) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
52 bnj964.5 . . . . 5 (𝜓′[𝑝 / 𝑛]𝜓)
53 vex 3496 . . . . 5 𝑝 ∈ V
542, 52, 53bnj539 32151 . . . 4 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑝 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
55 bnj964.8 . . . 4 (𝜓″[𝐺 / 𝑓]𝜓′)
56 bnj964.12 . . . 4 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
57 bnj964.13 . . . 4 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
5854, 55, 56, 57bnj965 32202 . . 3 (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑝 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
5958bnj115 31983 . 2 (𝜓″ ↔ ∀𝑖((𝑖 ∈ ω ∧ suc 𝑖𝑝) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
6051, 59sylibr 236 1 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜓″)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1081  ∀wal 1528   = wceq 1530   ∈ wcel 2107  ∀wral 3136  [wsbc 3770   ∪ cun 3932  {csn 4559  ⟨cop 4565  ∪ ciun 4910  suc csuc 6186   Fn wfn 6343  ‘cfv 6348  ωcom 7572   ∧ w-bnj17 31944   predc-bnj14 31946   FrSe w-bnj15 31950 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-suc 6190  df-iota 6307  df-fv 6356  df-bnj17 31945 This theorem is referenced by:  bnj910  32208
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