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Theorem bnj964 30756
Description: Technical lemma for bnj69 30821. 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 1840 . . . 4 𝑖(𝑅 FrSe 𝐴𝑋𝐴)
2 bnj964.2 . . . . . . . 8 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
32bnj1095 30595 . . . . . . 7 (𝜓 → ∀𝑖𝜓)
4 bnj964.3 . . . . . . 7 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
53, 4bnj1096 30596 . . . . . 6 (𝜒 → ∀𝑖𝜒)
65nf5i 2021 . . . . 5 𝑖𝜒
7 nfv 1840 . . . . 5 𝑖 𝑛 = suc 𝑚
8 nfv 1840 . . . . 5 𝑖 𝑝 = suc 𝑛
96, 7, 8nf3an 1828 . . . 4 𝑖(𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)
101, 9nfan 1825 . . 3 𝑖((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛))
11 bnj255 30513 . . . . 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 30563 . . . . . . 7 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝) → suc 𝑖𝑝)
13 simp3 1061 . . . . . . . 8 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝑝 = suc 𝑛)
1413bnj706 30567 . . . . . . 7 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝) → 𝑝 = suc 𝑛)
15 eleq2 2687 . . . . . . . . 9 (𝑝 = suc 𝑛 → (suc 𝑖𝑝 ↔ suc 𝑖 ∈ suc 𝑛))
1615biimpac 503 . . . . . . . 8 ((suc 𝑖𝑝𝑝 = suc 𝑛) → suc 𝑖 ∈ suc 𝑛)
17 elsuci 5755 . . . . . . . . 9 (suc 𝑖 ∈ suc 𝑛 → (suc 𝑖𝑛 ∨ suc 𝑖 = 𝑛))
18 eqcom 2628 . . . . . . . . . 10 (suc 𝑖 = 𝑛𝑛 = suc 𝑖)
1918orbi2i 541 . . . . . . . . 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 692 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖𝑝) → (suc 𝑖𝑛𝑛 = suc 𝑖))
23 df-3an 1038 . . . . . . . . . . . . 13 ((𝑖 ∈ ω ∧ suc 𝑖𝑝 ∧ suc 𝑖𝑛) ↔ ((𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ suc 𝑖𝑛))
24233anbi3i 1253 . . . . . . . . . . . 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 30513 . . . . . . . . . . . 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 30522 . . . . . . . . . . 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 30511 . . . . . . . . . . 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 729 . . . . . . . . . 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 450 . . . . . . 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 1038 . . . . . . . . . . . . 13 ((𝑖 ∈ ω ∧ suc 𝑖𝑝𝑛 = suc 𝑖) ↔ ((𝑖 ∈ ω ∧ suc 𝑖𝑝) ∧ 𝑛 = suc 𝑖))
36353anbi3i 1253 . . . . . . . . . . . 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 30513 . . . . . . . . . . . 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 30522 . . . . . . . . . . 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 30511 . . . . . . . . . . 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 729 . . . . . . . . . 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 450 . . . . . . 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 394 . . . . . 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 1264 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → ((𝑖 ∈ ω ∧ suc 𝑖𝑝) → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
5110, 50alrimi 2080 . 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 3192 . . . . 5 𝑝 ∈ V
542, 52, 53bnj539 30704 . . . 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 30755 . . 3 (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑝 → (𝐺‘suc 𝑖) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)))
5958bnj115 30534 . 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  wo 383  wa 384  w3a 1036  wal 1478   = wceq 1480  wcel 1987  wral 2907  [wsbc 3421  cun 3557  {csn 4153  cop 4159   ciun 4490  suc csuc 5689   Fn wfn 5847  cfv 5852  ωcom 7019  w-bnj17 30494   predc-bnj14 30496   FrSe w-bnj15 30500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-suc 5693  df-iota 5815  df-fv 5860  df-bnj17 30495
This theorem is referenced by:  bnj910  30761
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