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Theorem bnj964 31351
Description: Technical lemma for bnj69 31416. 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 1995 . . . 4 𝑖(𝑅 FrSe 𝐴𝑋𝐴)
2 bnj964.2 . . . . . . . 8 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
32bnj1095 31190 . . . . . . 7 (𝜓 → ∀𝑖𝜓)
4 bnj964.3 . . . . . . 7 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
53, 4bnj1096 31191 . . . . . 6 (𝜒 → ∀𝑖𝜒)
65nf5i 2179 . . . . 5 𝑖𝜒
7 nfv 1995 . . . . 5 𝑖 𝑛 = suc 𝑚
8 nfv 1995 . . . . 5 𝑖 𝑝 = suc 𝑛
96, 7, 8nf3an 1983 . . . 4 𝑖(𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)
101, 9nfan 1980 . . 3 𝑖((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛))
11 bnj255 31111 . . . . 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 31158 . . . . . . 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 31162 . . . . . . 7 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝) → 𝑝 = suc 𝑛)
15 eleq2 2839 . . . . . . . . 9 (𝑝 = suc 𝑛 → (suc 𝑖𝑝 ↔ suc 𝑖 ∈ suc 𝑛))
1615biimpac 464 . . . . . . . 8 ((suc 𝑖𝑝𝑝 = suc 𝑛) → suc 𝑖 ∈ suc 𝑛)
17 elsuci 5933 . . . . . . . . 9 (suc 𝑖 ∈ suc 𝑛 → (suc 𝑖𝑛 ∨ suc 𝑖 = 𝑛))
18 eqcom 2778 . . . . . . . . . 10 (suc 𝑖 = 𝑛𝑛 = suc 𝑖)
1918orbi2i 898 . . . . . . . . 9 ((suc 𝑖𝑛 ∨ suc 𝑖 = 𝑛) ↔ (suc 𝑖𝑛𝑛 = suc 𝑖))
2017, 19sylib 208 . . . . . . . 8 (suc 𝑖 ∈ suc 𝑛 → (suc 𝑖𝑛𝑛 = suc 𝑖))
2116, 20syl 17 . . . . . . 7 ((suc 𝑖𝑝𝑝 = suc 𝑛) → (suc 𝑖𝑛𝑛 = suc 𝑖))
2212, 14, 21syl2anc 573 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝) → (suc 𝑖𝑛𝑛 = suc 𝑖))
23 df-3an 1073 . . . . . . . . . . . . 13 ((𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛) ↔ ((𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ suc 𝑖𝑛))
24233anbi3i 1162 . . . . . . . . . . . 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 31111 . . . . . . . . . . . 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 267 . . . . . . . . . . 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 31120 . . . . . . . . . . 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 31109 . . . . . . . . . . 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 286 . . . . . . . . . 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 609 . . . . . . . . . 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 267 . . . . . . . . 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 225 . . . . . . . 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 397 . . . . . . 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 1073 . . . . . . . . . . . . 13 ((𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖) ↔ ((𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ 𝑛 = suc 𝑖))
36353anbi3i 1162 . . . . . . . . . . . 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 31111 . . . . . . . . . . . 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 267 . . . . . . . . . . 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 31120 . . . . . . . . . . 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 31109 . . . . . . . . . . 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 286 . . . . . . . . . 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 609 . . . . . . . . . 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 267 . . . . . . . . 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 225 . . . . . . . 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 397 . . . . . . 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 846 . . . . . 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 225 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖𝑝)) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
50493expia 1114 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → ((𝑖 ∈ ω ∧ suc 𝑖𝑝) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
5110, 50alrimi 2238 . 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 3354 . . . . 5 𝑝 ∈ V
542, 52, 53bnj539 31299 . . . 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 31350 . . 3 (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑝 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
5958bnj115 31131 . 2 (𝜓″ ↔ ∀𝑖((𝑖 ∈ ω ∧ suc 𝑖𝑝) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
6051, 59sylibr 224 1 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜓″)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wo 836  w3a 1071  wal 1629   = wceq 1631  wcel 2145  wral 3061  [wsbc 3587  cun 3721  {csn 4317  cop 4323   ciun 4655  suc csuc 5867   Fn wfn 6025  cfv 6030  ωcom 7216  w-bnj17 31092   predc-bnj14 31094   FrSe w-bnj15 31098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-suc 5871  df-iota 5993  df-fv 6038  df-bnj17 31093
This theorem is referenced by:  bnj910  31356
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