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Theorem bnj1128 31506
Description: Technical lemma for bnj69 31526. 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
bnj1128.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1128.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1128.3 𝐷 = (ω ∖ {∅})
bnj1128.4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj1128.5 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1128.6 (𝜃 ↔ (𝜒 → (𝑓𝑖) ⊆ 𝐴))
bnj1128.7 (𝜏 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜃))
bnj1128.8 (𝜑′[𝑗 / 𝑖]𝜑)
bnj1128.9 (𝜓′[𝑗 / 𝑖]𝜓)
bnj1128.10 (𝜒′[𝑗 / 𝑖]𝜒)
bnj1128.11 (𝜃′[𝑗 / 𝑖]𝜃)
Assertion
Ref Expression
bnj1128 (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑌𝐴)
Distinct variable groups:   𝐴,𝑓,𝑖,𝑗,𝑛,𝑦   𝐷,𝑖,𝑗,𝑦   𝑅,𝑓,𝑖,𝑗,𝑛,𝑦   𝑓,𝑋,𝑖,𝑛,𝑦   𝑓,𝑌,𝑖,𝑛,𝑦   𝜒,𝑗   𝜑,𝑖,𝑦   𝜃,𝑗
Allowed substitution hints:   𝜑(𝑓,𝑗,𝑛)   𝜓(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜒(𝑦,𝑓,𝑖,𝑛)   𝜃(𝑦,𝑓,𝑖,𝑛)   𝜏(𝑦,𝑓,𝑖,𝑗,𝑛)   𝐵(𝑦,𝑓,𝑖,𝑗,𝑛)   𝐷(𝑓,𝑛)   𝑋(𝑗)   𝑌(𝑗)   𝜑′(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜓′(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜒′(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜃′(𝑦,𝑓,𝑖,𝑗,𝑛)

Proof of Theorem bnj1128
StepHypRef Expression
1 bnj1128.1 . . . 4 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2 bnj1128.2 . . . 4 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 bnj1128.3 . . . 4 𝐷 = (ω ∖ {∅})
4 bnj1128.4 . . . 4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
5 bnj1128.5 . . . 4 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
61, 2, 3, 4, 5bnj981 31468 . . 3 (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑌 ∈ (𝑓𝑖)))
7 simp1 1166 . . . . . 6 ((𝜒𝑖𝑛𝑌 ∈ (𝑓𝑖)) → 𝜒)
8 simp2 1167 . . . . . 6 ((𝜒𝑖𝑛𝑌 ∈ (𝑓𝑖)) → 𝑖𝑛)
9 bnj1128.7 . . . . . . . . 9 (𝜏 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜃))
10 nfv 2009 . . . . . . . . . . . . . . 15 𝑗 𝑖𝑛
11 nfra1 3088 . . . . . . . . . . . . . . . 16 𝑗𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜃)
129, 11nfxfr 1948 . . . . . . . . . . . . . . 15 𝑗𝜏
13 nfv 2009 . . . . . . . . . . . . . . 15 𝑗𝜒
1410, 12, 13nf3an 2000 . . . . . . . . . . . . . 14 𝑗(𝑖𝑛𝜏𝜒)
15 nfv 2009 . . . . . . . . . . . . . 14 𝑗(𝑓𝑖) ⊆ 𝐴
1614, 15nfim 1995 . . . . . . . . . . . . 13 𝑗((𝑖𝑛𝜏𝜒) → (𝑓𝑖) ⊆ 𝐴)
1716nf5ri 2227 . . . . . . . . . . . 12 (((𝑖𝑛𝜏𝜒) → (𝑓𝑖) ⊆ 𝐴) → ∀𝑗((𝑖𝑛𝜏𝜒) → (𝑓𝑖) ⊆ 𝐴))
183bnj1098 31302 . . . . . . . . . . . . . . . . 17 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗𝑛𝑖 = suc 𝑗))
19 simpl 474 . . . . . . . . . . . . . . . . . 18 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → 𝑖 ≠ ∅)
20 simpr1 1248 . . . . . . . . . . . . . . . . . 18 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → 𝑖𝑛)
215bnj1232 31322 . . . . . . . . . . . . . . . . . . . 20 (𝜒𝑛𝐷)
22213ad2ant3 1165 . . . . . . . . . . . . . . . . . . 19 ((𝑖𝑛𝜏𝜒) → 𝑛𝐷)
2322adantl 473 . . . . . . . . . . . . . . . . . 18 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → 𝑛𝐷)
2419, 20, 233jca 1158 . . . . . . . . . . . . . . . . 17 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷))
2518, 24bnj1101 31303 . . . . . . . . . . . . . . . 16 𝑗((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → (𝑗𝑛𝑖 = suc 𝑗))
26 ancl 540 . . . . . . . . . . . . . . . 16 (((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → (𝑗𝑛𝑖 = suc 𝑗)) → ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) ∧ (𝑗𝑛𝑖 = suc 𝑗))))
2725, 26bnj101 31240 . . . . . . . . . . . . . . 15 𝑗((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) ∧ (𝑗𝑛𝑖 = suc 𝑗)))
28 df-3an 1109 . . . . . . . . . . . . . . . . 17 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) ↔ ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) ∧ (𝑗𝑛𝑖 = suc 𝑗)))
2928imbi2i 327 . . . . . . . . . . . . . . . 16 (((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → (𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗))) ↔ ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) ∧ (𝑗𝑛𝑖 = suc 𝑗))))
3029exbii 1943 . . . . . . . . . . . . . . 15 (∃𝑗((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → (𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗))) ↔ ∃𝑗((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) ∧ (𝑗𝑛𝑖 = suc 𝑗))))
3127, 30mpbir 222 . . . . . . . . . . . . . 14 𝑗((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → (𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)))
32 bnj213 31400 . . . . . . . . . . . . . . . 16 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
3332bnj226 31251 . . . . . . . . . . . . . . 15 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
34 simp21 1263 . . . . . . . . . . . . . . . 16 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → 𝑖𝑛)
35 simp3r 1259 . . . . . . . . . . . . . . . . 17 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → 𝑖 = suc 𝑗)
36 biid 252 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛𝐷𝑛𝐷)
37 biid 252 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 Fn 𝑛𝑓 Fn 𝑛)
38 bnj1128.8 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑′[𝑗 / 𝑖]𝜑)
39 vex 3353 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑗 ∈ V
40 sbcg 3662 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ V → ([𝑗 / 𝑖]𝜑𝜑))
4139, 40ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 ([𝑗 / 𝑖]𝜑𝜑)
4238, 41bitr2i 267 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝜑′)
43 bnj1128.9 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜓′[𝑗 / 𝑖]𝜓)
442, 43bnj1039 31487 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
452, 44bitr4i 269 . . . . . . . . . . . . . . . . . . . . . 22 (𝜓𝜓′)
4636, 37, 42, 45bnj887 31283 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑′𝜓′))
47 bnj1128.10 . . . . . . . . . . . . . . . . . . . . . 22 (𝜒′[𝑗 / 𝑖]𝜒)
4838, 43, 5, 47bnj1040 31488 . . . . . . . . . . . . . . . . . . . . 21 (𝜒′ ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑′𝜓′))
4946, 5, 483bitr4i 294 . . . . . . . . . . . . . . . . . . . 20 (𝜒𝜒′)
5048bnj1254 31328 . . . . . . . . . . . . . . . . . . . 20 (𝜒′𝜓′)
5149, 50sylbi 208 . . . . . . . . . . . . . . . . . . 19 (𝜒𝜓′)
52513ad2ant3 1165 . . . . . . . . . . . . . . . . . 18 ((𝑖𝑛𝜏𝜒) → 𝜓′)
53523ad2ant2 1164 . . . . . . . . . . . . . . . . 17 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → 𝜓′)
54 simp3l 1258 . . . . . . . . . . . . . . . . . 18 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → 𝑗𝑛)
55223ad2ant2 1164 . . . . . . . . . . . . . . . . . 18 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → 𝑛𝐷)
563bnj923 31286 . . . . . . . . . . . . . . . . . . 19 (𝑛𝐷𝑛 ∈ ω)
57 elnn 7273 . . . . . . . . . . . . . . . . . . 19 ((𝑗𝑛𝑛 ∈ ω) → 𝑗 ∈ ω)
5856, 57sylan2 586 . . . . . . . . . . . . . . . . . 18 ((𝑗𝑛𝑛𝐷) → 𝑗 ∈ ω)
5954, 55, 58syl2anc 579 . . . . . . . . . . . . . . . . 17 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → 𝑗 ∈ ω)
6044bnj589 31427 . . . . . . . . . . . . . . . . . . 19 (𝜓′ ↔ ∀𝑗 ∈ ω (suc 𝑗𝑛 → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
61 rsp 3076 . . . . . . . . . . . . . . . . . . 19 (∀𝑗 ∈ ω (suc 𝑗𝑛 → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)) → (𝑗 ∈ ω → (suc 𝑗𝑛 → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))))
6260, 61sylbi 208 . . . . . . . . . . . . . . . . . 18 (𝜓′ → (𝑗 ∈ ω → (suc 𝑗𝑛 → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))))
63 eleq1 2832 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = suc 𝑗 → (𝑖𝑛 ↔ suc 𝑗𝑛))
64 fveqeq2 6384 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = suc 𝑗 → ((𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
6563, 64imbi12d 335 . . . . . . . . . . . . . . . . . . 19 (𝑖 = suc 𝑗 → ((𝑖𝑛 → (𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑗𝑛 → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))))
6665imbi2d 331 . . . . . . . . . . . . . . . . . 18 (𝑖 = suc 𝑗 → ((𝑗 ∈ ω → (𝑖𝑛 → (𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))) ↔ (𝑗 ∈ ω → (suc 𝑗𝑛 → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))))
6762, 66syl5ibr 237 . . . . . . . . . . . . . . . . 17 (𝑖 = suc 𝑗 → (𝜓′ → (𝑗 ∈ ω → (𝑖𝑛 → (𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))))
6835, 53, 59, 67syl3c 66 . . . . . . . . . . . . . . . 16 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → (𝑖𝑛 → (𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
6934, 68mpd 15 . . . . . . . . . . . . . . 15 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → (𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))
7033, 69bnj1262 31329 . . . . . . . . . . . . . 14 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → (𝑓𝑖) ⊆ 𝐴)
7131, 70bnj1023 31299 . . . . . . . . . . . . 13 𝑗((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → (𝑓𝑖) ⊆ 𝐴)
725bnj1247 31327 . . . . . . . . . . . . . . 15 (𝜒𝜑)
73723ad2ant3 1165 . . . . . . . . . . . . . 14 ((𝑖𝑛𝜏𝜒) → 𝜑)
74 bnj213 31400 . . . . . . . . . . . . . . 15 pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴
75 fveq2 6375 . . . . . . . . . . . . . . . 16 (𝑖 = ∅ → (𝑓𝑖) = (𝑓‘∅))
761biimpi 207 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
7775, 76sylan9eq 2819 . . . . . . . . . . . . . . 15 ((𝑖 = ∅ ∧ 𝜑) → (𝑓𝑖) = pred(𝑋, 𝐴, 𝑅))
7874, 77bnj1262 31329 . . . . . . . . . . . . . 14 ((𝑖 = ∅ ∧ 𝜑) → (𝑓𝑖) ⊆ 𝐴)
7973, 78sylan2 586 . . . . . . . . . . . . 13 ((𝑖 = ∅ ∧ (𝑖𝑛𝜏𝜒)) → (𝑓𝑖) ⊆ 𝐴)
8071, 79bnj1109 31305 . . . . . . . . . . . 12 𝑗((𝑖𝑛𝜏𝜒) → (𝑓𝑖) ⊆ 𝐴)
8117, 80bnj1131 31306 . . . . . . . . . . 11 ((𝑖𝑛𝜏𝜒) → (𝑓𝑖) ⊆ 𝐴)
82813expia 1150 . . . . . . . . . 10 ((𝑖𝑛𝜏) → (𝜒 → (𝑓𝑖) ⊆ 𝐴))
83 bnj1128.6 . . . . . . . . . 10 (𝜃 ↔ (𝜒 → (𝑓𝑖) ⊆ 𝐴))
8482, 83sylibr 225 . . . . . . . . 9 ((𝑖𝑛𝜏) → 𝜃)
853, 5, 9, 84bnj1133 31505 . . . . . . . 8 (𝜒 → ∀𝑖𝑛 𝜃)
8683ralbii 3127 . . . . . . . 8 (∀𝑖𝑛 𝜃 ↔ ∀𝑖𝑛 (𝜒 → (𝑓𝑖) ⊆ 𝐴))
8785, 86sylib 209 . . . . . . 7 (𝜒 → ∀𝑖𝑛 (𝜒 → (𝑓𝑖) ⊆ 𝐴))
88 rsp 3076 . . . . . . 7 (∀𝑖𝑛 (𝜒 → (𝑓𝑖) ⊆ 𝐴) → (𝑖𝑛 → (𝜒 → (𝑓𝑖) ⊆ 𝐴)))
8987, 88syl 17 . . . . . 6 (𝜒 → (𝑖𝑛 → (𝜒 → (𝑓𝑖) ⊆ 𝐴)))
907, 8, 7, 89syl3c 66 . . . . 5 ((𝜒𝑖𝑛𝑌 ∈ (𝑓𝑖)) → (𝑓𝑖) ⊆ 𝐴)
91 simp3 1168 . . . . 5 ((𝜒𝑖𝑛𝑌 ∈ (𝑓𝑖)) → 𝑌 ∈ (𝑓𝑖))
9290, 91sseldd 3762 . . . 4 ((𝜒𝑖𝑛𝑌 ∈ (𝑓𝑖)) → 𝑌𝐴)
93922eximi 1930 . . 3 (∃𝑛𝑖(𝜒𝑖𝑛𝑌 ∈ (𝑓𝑖)) → ∃𝑛𝑖 𝑌𝐴)
946, 93bnj593 31263 . 2 (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖 𝑌𝐴)
95 19.9v 2078 . . 3 (∃𝑓𝑛𝑖 𝑌𝐴 ↔ ∃𝑛𝑖 𝑌𝐴)
96 19.9v 2078 . . 3 (∃𝑛𝑖 𝑌𝐴 ↔ ∃𝑖 𝑌𝐴)
97 19.9v 2078 . . 3 (∃𝑖 𝑌𝐴𝑌𝐴)
9895, 96, 973bitri 288 . 2 (∃𝑓𝑛𝑖 𝑌𝐴𝑌𝐴)
9994, 98sylib 209 1 (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑌𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  {cab 2751  wne 2937  wral 3055  wrex 3056  Vcvv 3350  [wsbc 3596  cdif 3729  wss 3732  c0 4079  {csn 4334   ciun 4676   class class class wbr 4809   E cep 5189  suc csuc 5910   Fn wfn 6063  cfv 6068  ωcom 7263  w-bnj17 31203   predc-bnj14 31205   trClc-bnj18 31211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-tr 4912  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fn 6071  df-fv 6076  df-om 7264  df-bnj17 31204  df-bnj14 31206  df-bnj18 31212
This theorem is referenced by:  bnj1127  31507
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