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Theorem bnj1128 31386
Description: Technical lemma for bnj69 31406. 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 31348 . . 3 (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑌 ∈ (𝑓𝑖)))
7 simp1 1131 . . . . . 6 ((𝜒𝑖𝑛𝑌 ∈ (𝑓𝑖)) → 𝜒)
8 simp2 1132 . . . . . 6 ((𝜒𝑖𝑛𝑌 ∈ (𝑓𝑖)) → 𝑖𝑛)
9 bnj1128.7 . . . . . . . . 9 (𝜏 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜃))
10 nfv 1992 . . . . . . . . . . . . . . 15 𝑗 𝑖𝑛
11 nfra1 3079 . . . . . . . . . . . . . . . 16 𝑗𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜃)
129, 11nfxfr 1928 . . . . . . . . . . . . . . 15 𝑗𝜏
13 nfv 1992 . . . . . . . . . . . . . . 15 𝑗𝜒
1410, 12, 13nf3an 1980 . . . . . . . . . . . . . 14 𝑗(𝑖𝑛𝜏𝜒)
15 nfv 1992 . . . . . . . . . . . . . 14 𝑗(𝑓𝑖) ⊆ 𝐴
1614, 15nfim 1974 . . . . . . . . . . . . 13 𝑗((𝑖𝑛𝜏𝜒) → (𝑓𝑖) ⊆ 𝐴)
1716nf5ri 2212 . . . . . . . . . . . 12 (((𝑖𝑛𝜏𝜒) → (𝑓𝑖) ⊆ 𝐴) → ∀𝑗((𝑖𝑛𝜏𝜒) → (𝑓𝑖) ⊆ 𝐴))
183bnj1098 31182 . . . . . . . . . . . . . . . . 17 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗𝑛𝑖 = suc 𝑗))
19 simpl 474 . . . . . . . . . . . . . . . . . 18 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → 𝑖 ≠ ∅)
20 simpr1 1234 . . . . . . . . . . . . . . . . . 18 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → 𝑖𝑛)
215bnj1232 31202 . . . . . . . . . . . . . . . . . . . 20 (𝜒𝑛𝐷)
22213ad2ant3 1130 . . . . . . . . . . . . . . . . . . 19 ((𝑖𝑛𝜏𝜒) → 𝑛𝐷)
2322adantl 473 . . . . . . . . . . . . . . . . . 18 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → 𝑛𝐷)
2419, 20, 233jca 1123 . . . . . . . . . . . . . . . . 17 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷))
2518, 24bnj1101 31183 . . . . . . . . . . . . . . . 16 𝑗((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → (𝑗𝑛𝑖 = suc 𝑗))
26 ancl 570 . . . . . . . . . . . . . . . 16 (((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → (𝑗𝑛𝑖 = suc 𝑗)) → ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) ∧ (𝑗𝑛𝑖 = suc 𝑗))))
2725, 26bnj101 31119 . . . . . . . . . . . . . . 15 𝑗((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) ∧ (𝑗𝑛𝑖 = suc 𝑗)))
28 df-3an 1074 . . . . . . . . . . . . . . . . 17 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) ↔ ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) ∧ (𝑗𝑛𝑖 = suc 𝑗)))
2928imbi2i 325 . . . . . . . . . . . . . . . 16 (((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → (𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗))) ↔ ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) ∧ (𝑗𝑛𝑖 = suc 𝑗))))
3029exbii 1923 . . . . . . . . . . . . . . 15 (∃𝑗((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → (𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗))) ↔ ∃𝑗((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) ∧ (𝑗𝑛𝑖 = suc 𝑗))))
3127, 30mpbir 221 . . . . . . . . . . . . . 14 𝑗((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → (𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)))
32 bnj213 31280 . . . . . . . . . . . . . . . 16 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
3332bnj226 31130 . . . . . . . . . . . . . . 15 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
34 simp21 1249 . . . . . . . . . . . . . . . 16 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → 𝑖𝑛)
35 simp3r 1245 . . . . . . . . . . . . . . . . 17 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → 𝑖 = suc 𝑗)
36 biid 251 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛𝐷𝑛𝐷)
37 biid 251 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 Fn 𝑛𝑓 Fn 𝑛)
38 bnj1128.8 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑′[𝑗 / 𝑖]𝜑)
39 vex 3343 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑗 ∈ V
40 sbcg 3644 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ V → ([𝑗 / 𝑖]𝜑𝜑))
4139, 40ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 ([𝑗 / 𝑖]𝜑𝜑)
4238, 41bitr2i 265 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝜑′)
43 bnj1128.9 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜓′[𝑗 / 𝑖]𝜓)
442, 43bnj1039 31367 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
452, 44bitr4i 267 . . . . . . . . . . . . . . . . . . . . . 22 (𝜓𝜓′)
4636, 37, 42, 45bnj887 31163 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑′𝜓′))
47 bnj1128.10 . . . . . . . . . . . . . . . . . . . . . 22 (𝜒′[𝑗 / 𝑖]𝜒)
4838, 43, 5, 47bnj1040 31368 . . . . . . . . . . . . . . . . . . . . 21 (𝜒′ ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑′𝜓′))
4946, 5, 483bitr4i 292 . . . . . . . . . . . . . . . . . . . 20 (𝜒𝜒′)
5048bnj1254 31208 . . . . . . . . . . . . . . . . . . . 20 (𝜒′𝜓′)
5149, 50sylbi 207 . . . . . . . . . . . . . . . . . . 19 (𝜒𝜓′)
52513ad2ant3 1130 . . . . . . . . . . . . . . . . . 18 ((𝑖𝑛𝜏𝜒) → 𝜓′)
53523ad2ant2 1129 . . . . . . . . . . . . . . . . 17 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → 𝜓′)
54 simp3l 1244 . . . . . . . . . . . . . . . . . 18 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → 𝑗𝑛)
55223ad2ant2 1129 . . . . . . . . . . . . . . . . . 18 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → 𝑛𝐷)
563bnj923 31166 . . . . . . . . . . . . . . . . . . 19 (𝑛𝐷𝑛 ∈ ω)
57 elnn 7241 . . . . . . . . . . . . . . . . . . 19 ((𝑗𝑛𝑛 ∈ ω) → 𝑗 ∈ ω)
5856, 57sylan2 492 . . . . . . . . . . . . . . . . . 18 ((𝑗𝑛𝑛𝐷) → 𝑗 ∈ ω)
5954, 55, 58syl2anc 696 . . . . . . . . . . . . . . . . 17 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → 𝑗 ∈ ω)
6044bnj589 31307 . . . . . . . . . . . . . . . . . . 19 (𝜓′ ↔ ∀𝑗 ∈ ω (suc 𝑗𝑛 → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
61 rsp 3067 . . . . . . . . . . . . . . . . . . 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 207 . . . . . . . . . . . . . . . . . 18 (𝜓′ → (𝑗 ∈ ω → (suc 𝑗𝑛 → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))))
63 eleq1 2827 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = suc 𝑗 → (𝑖𝑛 ↔ suc 𝑗𝑛))
64 fveq2 6353 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = suc 𝑗 → (𝑓𝑖) = (𝑓‘suc 𝑗))
6564eqeq1d 2762 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = suc 𝑗 → ((𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
6663, 65imbi12d 333 . . . . . . . . . . . . . . . . . . 19 (𝑖 = suc 𝑗 → ((𝑖𝑛 → (𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑗𝑛 → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))))
6766imbi2d 329 . . . . . . . . . . . . . . . . . 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)))))
6862, 67syl5ibr 236 . . . . . . . . . . . . . . . . 17 (𝑖 = suc 𝑗 → (𝜓′ → (𝑗 ∈ ω → (𝑖𝑛 → (𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))))
6935, 53, 59, 68syl3c 66 . . . . . . . . . . . . . . . 16 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → (𝑖𝑛 → (𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
7034, 69mpd 15 . . . . . . . . . . . . . . 15 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → (𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))
7133, 70bnj1262 31209 . . . . . . . . . . . . . 14 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → (𝑓𝑖) ⊆ 𝐴)
7231, 71bnj1023 31179 . . . . . . . . . . . . 13 𝑗((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → (𝑓𝑖) ⊆ 𝐴)
735bnj1247 31207 . . . . . . . . . . . . . . 15 (𝜒𝜑)
74733ad2ant3 1130 . . . . . . . . . . . . . 14 ((𝑖𝑛𝜏𝜒) → 𝜑)
75 bnj213 31280 . . . . . . . . . . . . . . 15 pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴
76 fveq2 6353 . . . . . . . . . . . . . . . 16 (𝑖 = ∅ → (𝑓𝑖) = (𝑓‘∅))
771biimpi 206 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
7876, 77sylan9eq 2814 . . . . . . . . . . . . . . 15 ((𝑖 = ∅ ∧ 𝜑) → (𝑓𝑖) = pred(𝑋, 𝐴, 𝑅))
7975, 78bnj1262 31209 . . . . . . . . . . . . . 14 ((𝑖 = ∅ ∧ 𝜑) → (𝑓𝑖) ⊆ 𝐴)
8074, 79sylan2 492 . . . . . . . . . . . . 13 ((𝑖 = ∅ ∧ (𝑖𝑛𝜏𝜒)) → (𝑓𝑖) ⊆ 𝐴)
8172, 80bnj1109 31185 . . . . . . . . . . . 12 𝑗((𝑖𝑛𝜏𝜒) → (𝑓𝑖) ⊆ 𝐴)
8217, 81bnj1131 31186 . . . . . . . . . . 11 ((𝑖𝑛𝜏𝜒) → (𝑓𝑖) ⊆ 𝐴)
83823expia 1115 . . . . . . . . . 10 ((𝑖𝑛𝜏) → (𝜒 → (𝑓𝑖) ⊆ 𝐴))
84 bnj1128.6 . . . . . . . . . 10 (𝜃 ↔ (𝜒 → (𝑓𝑖) ⊆ 𝐴))
8583, 84sylibr 224 . . . . . . . . 9 ((𝑖𝑛𝜏) → 𝜃)
863, 5, 9, 85bnj1133 31385 . . . . . . . 8 (𝜒 → ∀𝑖𝑛 𝜃)
8784ralbii 3118 . . . . . . . 8 (∀𝑖𝑛 𝜃 ↔ ∀𝑖𝑛 (𝜒 → (𝑓𝑖) ⊆ 𝐴))
8886, 87sylib 208 . . . . . . 7 (𝜒 → ∀𝑖𝑛 (𝜒 → (𝑓𝑖) ⊆ 𝐴))
89 rsp 3067 . . . . . . 7 (∀𝑖𝑛 (𝜒 → (𝑓𝑖) ⊆ 𝐴) → (𝑖𝑛 → (𝜒 → (𝑓𝑖) ⊆ 𝐴)))
9088, 89syl 17 . . . . . 6 (𝜒 → (𝑖𝑛 → (𝜒 → (𝑓𝑖) ⊆ 𝐴)))
917, 8, 7, 90syl3c 66 . . . . 5 ((𝜒𝑖𝑛𝑌 ∈ (𝑓𝑖)) → (𝑓𝑖) ⊆ 𝐴)
92 simp3 1133 . . . . 5 ((𝜒𝑖𝑛𝑌 ∈ (𝑓𝑖)) → 𝑌 ∈ (𝑓𝑖))
9391, 92sseldd 3745 . . . 4 ((𝜒𝑖𝑛𝑌 ∈ (𝑓𝑖)) → 𝑌𝐴)
94932eximi 1912 . . 3 (∃𝑛𝑖(𝜒𝑖𝑛𝑌 ∈ (𝑓𝑖)) → ∃𝑛𝑖 𝑌𝐴)
956, 94bnj593 31143 . 2 (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖 𝑌𝐴)
96 19.9v 2062 . . 3 (∃𝑓𝑛𝑖 𝑌𝐴 ↔ ∃𝑛𝑖 𝑌𝐴)
97 19.9v 2062 . . 3 (∃𝑛𝑖 𝑌𝐴 ↔ ∃𝑖 𝑌𝐴)
98 19.9v 2062 . . 3 (∃𝑖 𝑌𝐴𝑌𝐴)
9996, 97, 983bitri 286 . 2 (∃𝑓𝑛𝑖 𝑌𝐴𝑌𝐴)
10095, 99sylib 208 1 (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑌𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wex 1853  wcel 2139  {cab 2746  wne 2932  wral 3050  wrex 3051  Vcvv 3340  [wsbc 3576  cdif 3712  wss 3715  c0 4058  {csn 4321   ciun 4672   class class class wbr 4804   E cep 5178  suc csuc 5886   Fn wfn 6044  cfv 6049  ωcom 7231  w-bnj17 31082   predc-bnj14 31084   trClc-bnj18 31090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-tr 4905  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fn 6052  df-fv 6057  df-om 7232  df-bnj17 31083  df-bnj14 31085  df-bnj18 31091
This theorem is referenced by:  bnj1127  31387
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