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Theorem bnj1098 29911
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1098.1 𝐷 = (ω ∖ {∅})
Assertion
Ref Expression
bnj1098 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗𝑛𝑖 = suc 𝑗))
Distinct variable groups:   𝐷,𝑗   𝑖,𝑗   𝑗,𝑛
Allowed substitution hints:   𝐷(𝑖,𝑛)

Proof of Theorem bnj1098
StepHypRef Expression
1 3anrev 1041 . . . . . . 7 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) ↔ (𝑛𝐷𝑖𝑛𝑖 ≠ ∅))
2 df-3an 1032 . . . . . . 7 ((𝑛𝐷𝑖𝑛𝑖 ≠ ∅) ↔ ((𝑛𝐷𝑖𝑛) ∧ 𝑖 ≠ ∅))
31, 2bitri 262 . . . . . 6 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) ↔ ((𝑛𝐷𝑖𝑛) ∧ 𝑖 ≠ ∅))
4 simpr 475 . . . . . . . 8 ((𝑛𝐷𝑖𝑛) → 𝑖𝑛)
5 bnj1098.1 . . . . . . . . . 10 𝐷 = (ω ∖ {∅})
65bnj923 29895 . . . . . . . . 9 (𝑛𝐷𝑛 ∈ ω)
76adantr 479 . . . . . . . 8 ((𝑛𝐷𝑖𝑛) → 𝑛 ∈ ω)
8 elnn 6941 . . . . . . . 8 ((𝑖𝑛𝑛 ∈ ω) → 𝑖 ∈ ω)
94, 7, 8syl2anc 690 . . . . . . 7 ((𝑛𝐷𝑖𝑛) → 𝑖 ∈ ω)
109anim1i 589 . . . . . 6 (((𝑛𝐷𝑖𝑛) ∧ 𝑖 ≠ ∅) → (𝑖 ∈ ω ∧ 𝑖 ≠ ∅))
113, 10sylbi 205 . . . . 5 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑖 ∈ ω ∧ 𝑖 ≠ ∅))
12 nnsuc 6948 . . . . 5 ((𝑖 ∈ ω ∧ 𝑖 ≠ ∅) → ∃𝑗 ∈ ω 𝑖 = suc 𝑗)
1311, 12syl 17 . . . 4 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ∃𝑗 ∈ ω 𝑖 = suc 𝑗)
14 df-rex 2898 . . . . . 6 (∃𝑗 ∈ ω 𝑖 = suc 𝑗 ↔ ∃𝑗(𝑗 ∈ ω ∧ 𝑖 = suc 𝑗))
1514imbi2i 324 . . . . 5 (((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ∃𝑗 ∈ ω 𝑖 = suc 𝑗) ↔ ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ∃𝑗(𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)))
16 19.37v 1896 . . . . 5 (∃𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)) ↔ ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ∃𝑗(𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)))
1715, 16bitr4i 265 . . . 4 (((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ∃𝑗 ∈ ω 𝑖 = suc 𝑗) ↔ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)))
1813, 17mpbi 218 . . 3 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗 ∈ ω ∧ 𝑖 = suc 𝑗))
19 ancr 569 . . 3 (((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)) → ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷))))
2018, 19bnj101 29846 . 2 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)))
21 vex 3172 . . . . . 6 𝑗 ∈ V
2221bnj216 29857 . . . . 5 (𝑖 = suc 𝑗𝑗𝑖)
2322ad2antlr 758 . . . 4 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → 𝑗𝑖)
24 simpr2 1060 . . . 4 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → 𝑖𝑛)
25 3simpc 1052 . . . . . . 7 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑖𝑛𝑛𝐷))
2625ancomd 465 . . . . . 6 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑛𝐷𝑖𝑛))
2726adantl 480 . . . . 5 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → (𝑛𝐷𝑖𝑛))
28 nnord 6939 . . . . 5 (𝑛 ∈ ω → Ord 𝑛)
29 ordtr1 5667 . . . . 5 (Ord 𝑛 → ((𝑗𝑖𝑖𝑛) → 𝑗𝑛))
3027, 7, 28, 294syl 19 . . . 4 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → ((𝑗𝑖𝑖𝑛) → 𝑗𝑛))
3123, 24, 30mp2and 710 . . 3 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → 𝑗𝑛)
32 simplr 787 . . 3 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → 𝑖 = suc 𝑗)
3331, 32jca 552 . 2 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → (𝑗𝑛𝑖 = suc 𝑗))
3420, 33bnj1023 29908 1 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗𝑛𝑖 = suc 𝑗))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wex 1694  wcel 1976  wne 2776  wrex 2893  cdif 3533  c0 3870  {csn 4121  Ord word 5622  suc csuc 5625  ωcom 6931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-tr 4672  df-eprel 4936  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-om 6932
This theorem is referenced by:  bnj1110  30107  bnj1128  30115  bnj1145  30118
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