Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1128 Structured version   Visualization version   GIF version

Theorem bnj1128 32347
Description: Technical lemma for bnj69 32367. 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 32307 . . 3 (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑌 ∈ (𝑓𝑖)))
7 simp1 1133 . . . . . 6 ((𝜒𝑖𝑛𝑌 ∈ (𝑓𝑖)) → 𝜒)
8 simp2 1134 . . . . . 6 ((𝜒𝑖𝑛𝑌 ∈ (𝑓𝑖)) → 𝑖𝑛)
9 bnj1128.7 . . . . . . . . 9 (𝜏 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜃))
10 nfv 1916 . . . . . . . . . . . . . . 15 𝑗 𝑖𝑛
11 nfra1 3213 . . . . . . . . . . . . . . . 16 𝑗𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜃)
129, 11nfxfr 1854 . . . . . . . . . . . . . . 15 𝑗𝜏
13 nfv 1916 . . . . . . . . . . . . . . 15 𝑗𝜒
1410, 12, 13nf3an 1903 . . . . . . . . . . . . . 14 𝑗(𝑖𝑛𝜏𝜒)
15 nfv 1916 . . . . . . . . . . . . . 14 𝑗(𝑓𝑖) ⊆ 𝐴
1614, 15nfim 1898 . . . . . . . . . . . . 13 𝑗((𝑖𝑛𝜏𝜒) → (𝑓𝑖) ⊆ 𝐴)
1716nf5ri 2197 . . . . . . . . . . . 12 (((𝑖𝑛𝜏𝜒) → (𝑓𝑖) ⊆ 𝐴) → ∀𝑗((𝑖𝑛𝜏𝜒) → (𝑓𝑖) ⊆ 𝐴))
183bnj1098 32140 . . . . . . . . . . . . . . . . 17 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗𝑛𝑖 = suc 𝑗))
19 simpl 486 . . . . . . . . . . . . . . . . . 18 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → 𝑖 ≠ ∅)
20 simpr1 1191 . . . . . . . . . . . . . . . . . 18 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → 𝑖𝑛)
215bnj1232 32160 . . . . . . . . . . . . . . . . . . . 20 (𝜒𝑛𝐷)
22213ad2ant3 1132 . . . . . . . . . . . . . . . . . . 19 ((𝑖𝑛𝜏𝜒) → 𝑛𝐷)
2322adantl 485 . . . . . . . . . . . . . . . . . 18 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → 𝑛𝐷)
2419, 20, 233jca 1125 . . . . . . . . . . . . . . . . 17 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷))
2518, 24bnj1101 32141 . . . . . . . . . . . . . . . 16 𝑗((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → (𝑗𝑛𝑖 = suc 𝑗))
26 ancl 548 . . . . . . . . . . . . . . . 16 (((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → (𝑗𝑛𝑖 = suc 𝑗)) → ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) ∧ (𝑗𝑛𝑖 = suc 𝑗))))
2725, 26bnj101 32078 . . . . . . . . . . . . . . 15 𝑗((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) ∧ (𝑗𝑛𝑖 = suc 𝑗)))
28 df-3an 1086 . . . . . . . . . . . . . . . . 17 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) ↔ ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) ∧ (𝑗𝑛𝑖 = suc 𝑗)))
2928imbi2i 339 . . . . . . . . . . . . . . . 16 (((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → (𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗))) ↔ ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) ∧ (𝑗𝑛𝑖 = suc 𝑗))))
3029exbii 1849 . . . . . . . . . . . . . . 15 (∃𝑗((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → (𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗))) ↔ ∃𝑗((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) ∧ (𝑗𝑛𝑖 = suc 𝑗))))
3127, 30mpbir 234 . . . . . . . . . . . . . 14 𝑗((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → (𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)))
32 bnj213 32239 . . . . . . . . . . . . . . . 16 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
3332bnj226 32089 . . . . . . . . . . . . . . 15 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
34 simp21 1203 . . . . . . . . . . . . . . . 16 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → 𝑖𝑛)
35 simp3r 1199 . . . . . . . . . . . . . . . . 17 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → 𝑖 = suc 𝑗)
36 biid 264 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛𝐷𝑛𝐷)
37 biid 264 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 Fn 𝑛𝑓 Fn 𝑛)
38 bnj1128.8 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑′[𝑗 / 𝑖]𝜑)
39 vex 3483 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑗 ∈ V
40 sbcg 3831 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ V → ([𝑗 / 𝑖]𝜑𝜑))
4139, 40ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 ([𝑗 / 𝑖]𝜑𝜑)
4238, 41bitr2i 279 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝜑′)
43 bnj1128.9 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜓′[𝑗 / 𝑖]𝜓)
442, 43bnj1039 32328 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
452, 44bitr4i 281 . . . . . . . . . . . . . . . . . . . . . 22 (𝜓𝜓′)
4636, 37, 42, 45bnj887 32121 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑′𝜓′))
47 bnj1128.10 . . . . . . . . . . . . . . . . . . . . . 22 (𝜒′[𝑗 / 𝑖]𝜒)
4838, 43, 5, 47bnj1040 32329 . . . . . . . . . . . . . . . . . . . . 21 (𝜒′ ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑′𝜓′))
4946, 5, 483bitr4i 306 . . . . . . . . . . . . . . . . . . . 20 (𝜒𝜒′)
5048bnj1254 32166 . . . . . . . . . . . . . . . . . . . 20 (𝜒′𝜓′)
5149, 50sylbi 220 . . . . . . . . . . . . . . . . . . 19 (𝜒𝜓′)
52513ad2ant3 1132 . . . . . . . . . . . . . . . . . 18 ((𝑖𝑛𝜏𝜒) → 𝜓′)
53523ad2ant2 1131 . . . . . . . . . . . . . . . . 17 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → 𝜓′)
54 simp3l 1198 . . . . . . . . . . . . . . . . . 18 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → 𝑗𝑛)
55223ad2ant2 1131 . . . . . . . . . . . . . . . . . 18 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → 𝑛𝐷)
563bnj923 32124 . . . . . . . . . . . . . . . . . . 19 (𝑛𝐷𝑛 ∈ ω)
57 elnn 7586 . . . . . . . . . . . . . . . . . . 19 ((𝑗𝑛𝑛 ∈ ω) → 𝑗 ∈ ω)
5856, 57sylan2 595 . . . . . . . . . . . . . . . . . 18 ((𝑗𝑛𝑛𝐷) → 𝑗 ∈ ω)
5954, 55, 58syl2anc 587 . . . . . . . . . . . . . . . . 17 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → 𝑗 ∈ ω)
6044bnj589 32266 . . . . . . . . . . . . . . . . . . 19 (𝜓′ ↔ ∀𝑗 ∈ ω (suc 𝑗𝑛 → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
61 rsp 3200 . . . . . . . . . . . . . . . . . . 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 220 . . . . . . . . . . . . . . . . . 18 (𝜓′ → (𝑗 ∈ ω → (suc 𝑗𝑛 → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))))
63 eleq1 2903 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = suc 𝑗 → (𝑖𝑛 ↔ suc 𝑗𝑛))
64 fveqeq2 6672 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = suc 𝑗 → ((𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
6563, 64imbi12d 348 . . . . . . . . . . . . . . . . . . 19 (𝑖 = suc 𝑗 → ((𝑖𝑛 → (𝑓𝑖) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑗𝑛 → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))))
6665imbi2d 344 . . . . . . . . . . . . . . . . . 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 249 . . . . . . . . . . . . . . . . 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 32167 . . . . . . . . . . . . . 14 ((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)) → (𝑓𝑖) ⊆ 𝐴)
7131, 70bnj1023 32137 . . . . . . . . . . . . 13 𝑗((𝑖 ≠ ∅ ∧ (𝑖𝑛𝜏𝜒)) → (𝑓𝑖) ⊆ 𝐴)
725bnj1247 32165 . . . . . . . . . . . . . . 15 (𝜒𝜑)
73723ad2ant3 1132 . . . . . . . . . . . . . 14 ((𝑖𝑛𝜏𝜒) → 𝜑)
74 bnj213 32239 . . . . . . . . . . . . . . 15 pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴
75 fveq2 6663 . . . . . . . . . . . . . . . 16 (𝑖 = ∅ → (𝑓𝑖) = (𝑓‘∅))
761biimpi 219 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
7775, 76sylan9eq 2879 . . . . . . . . . . . . . . 15 ((𝑖 = ∅ ∧ 𝜑) → (𝑓𝑖) = pred(𝑋, 𝐴, 𝑅))
7874, 77bnj1262 32167 . . . . . . . . . . . . . 14 ((𝑖 = ∅ ∧ 𝜑) → (𝑓𝑖) ⊆ 𝐴)
7973, 78sylan2 595 . . . . . . . . . . . . 13 ((𝑖 = ∅ ∧ (𝑖𝑛𝜏𝜒)) → (𝑓𝑖) ⊆ 𝐴)
8071, 79bnj1109 32143 . . . . . . . . . . . 12 𝑗((𝑖𝑛𝜏𝜒) → (𝑓𝑖) ⊆ 𝐴)
8117, 80bnj1131 32144 . . . . . . . . . . 11 ((𝑖𝑛𝜏𝜒) → (𝑓𝑖) ⊆ 𝐴)
82813expia 1118 . . . . . . . . . 10 ((𝑖𝑛𝜏) → (𝜒 → (𝑓𝑖) ⊆ 𝐴))
83 bnj1128.6 . . . . . . . . . 10 (𝜃 ↔ (𝜒 → (𝑓𝑖) ⊆ 𝐴))
8482, 83sylibr 237 . . . . . . . . 9 ((𝑖𝑛𝜏) → 𝜃)
853, 5, 9, 84bnj1133 32346 . . . . . . . 8 (𝜒 → ∀𝑖𝑛 𝜃)
8683ralbii 3160 . . . . . . . 8 (∀𝑖𝑛 𝜃 ↔ ∀𝑖𝑛 (𝜒 → (𝑓𝑖) ⊆ 𝐴))
8785, 86sylib 221 . . . . . . 7 (𝜒 → ∀𝑖𝑛 (𝜒 → (𝑓𝑖) ⊆ 𝐴))
88 rsp 3200 . . . . . . 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 1135 . . . . 5 ((𝜒𝑖𝑛𝑌 ∈ (𝑓𝑖)) → 𝑌 ∈ (𝑓𝑖))
9290, 91sseldd 3954 . . . 4 ((𝜒𝑖𝑛𝑌 ∈ (𝑓𝑖)) → 𝑌𝐴)
93922eximi 1837 . . 3 (∃𝑛𝑖(𝜒𝑖𝑛𝑌 ∈ (𝑓𝑖)) → ∃𝑛𝑖 𝑌𝐴)
946, 93bnj593 32101 . 2 (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖 𝑌𝐴)
95 19.9v 1989 . . 3 (∃𝑓𝑛𝑖 𝑌𝐴 ↔ ∃𝑛𝑖 𝑌𝐴)
96 19.9v 1989 . . 3 (∃𝑛𝑖 𝑌𝐴 ↔ ∃𝑖 𝑌𝐴)
97 19.9v 1989 . . 3 (∃𝑖 𝑌𝐴𝑌𝐴)
9895, 96, 973bitri 300 . 2 (∃𝑓𝑛𝑖 𝑌𝐴𝑌𝐴)
9994, 98sylib 221 1 (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑌𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2115  {cab 2802  wne 3014  wral 3133  wrex 3134  Vcvv 3480  [wsbc 3758  cdif 3916  wss 3919  c0 4276  {csn 4550   ciun 4905   class class class wbr 5053   E cep 5452  suc csuc 6182   Fn wfn 6340  cfv 6345  ωcom 7576  w-bnj17 32041   predc-bnj14 32043   trClc-bnj18 32049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318  ax-un 7457
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-tr 5160  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fn 6348  df-fv 6353  df-om 7577  df-bnj17 32042  df-bnj14 32044  df-bnj18 32050
This theorem is referenced by:  bnj1127  32348
  Copyright terms: Public domain W3C validator