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Theorem bnj1098 34409
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 1099 . . . . . . 7 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) ↔ (𝑛𝐷𝑖𝑛𝑖 ≠ ∅))
2 df-3an 1087 . . . . . . 7 ((𝑛𝐷𝑖𝑛𝑖 ≠ ∅) ↔ ((𝑛𝐷𝑖𝑛) ∧ 𝑖 ≠ ∅))
31, 2bitri 275 . . . . . 6 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) ↔ ((𝑛𝐷𝑖𝑛) ∧ 𝑖 ≠ ∅))
4 simpr 484 . . . . . . . 8 ((𝑛𝐷𝑖𝑛) → 𝑖𝑛)
5 bnj1098.1 . . . . . . . . . 10 𝐷 = (ω ∖ {∅})
65bnj923 34394 . . . . . . . . 9 (𝑛𝐷𝑛 ∈ ω)
76adantr 480 . . . . . . . 8 ((𝑛𝐷𝑖𝑛) → 𝑛 ∈ ω)
8 elnn 7876 . . . . . . . 8 ((𝑖𝑛𝑛 ∈ ω) → 𝑖 ∈ ω)
94, 7, 8syl2anc 583 . . . . . . 7 ((𝑛𝐷𝑖𝑛) → 𝑖 ∈ ω)
109anim1i 614 . . . . . 6 (((𝑛𝐷𝑖𝑛) ∧ 𝑖 ≠ ∅) → (𝑖 ∈ ω ∧ 𝑖 ≠ ∅))
113, 10sylbi 216 . . . . 5 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑖 ∈ ω ∧ 𝑖 ≠ ∅))
12 nnsuc 7883 . . . . 5 ((𝑖 ∈ ω ∧ 𝑖 ≠ ∅) → ∃𝑗 ∈ ω 𝑖 = suc 𝑗)
1311, 12syl 17 . . . 4 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ∃𝑗 ∈ ω 𝑖 = suc 𝑗)
14 df-rex 3067 . . . . . 6 (∃𝑗 ∈ ω 𝑖 = suc 𝑗 ↔ ∃𝑗(𝑗 ∈ ω ∧ 𝑖 = suc 𝑗))
1514imbi2i 336 . . . . 5 (((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ∃𝑗 ∈ ω 𝑖 = suc 𝑗) ↔ ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ∃𝑗(𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)))
16 19.37v 1988 . . . . 5 (∃𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)) ↔ ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ∃𝑗(𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)))
1715, 16bitr4i 278 . . . 4 (((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ∃𝑗 ∈ ω 𝑖 = suc 𝑗) ↔ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)))
1813, 17mpbi 229 . . 3 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗 ∈ ω ∧ 𝑖 = suc 𝑗))
19 ancr 546 . . 3 (((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)) → ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷))))
2018, 19bnj101 34349 . 2 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)))
21 vex 3474 . . . . . 6 𝑗 ∈ V
2221bnj216 34358 . . . . 5 (𝑖 = suc 𝑗𝑗𝑖)
2322ad2antlr 726 . . . 4 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → 𝑗𝑖)
24 simpr2 1193 . . . 4 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → 𝑖𝑛)
25 3simpc 1148 . . . . . . 7 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑖𝑛𝑛𝐷))
2625ancomd 461 . . . . . 6 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑛𝐷𝑖𝑛))
2726adantl 481 . . . . 5 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → (𝑛𝐷𝑖𝑛))
28 nnord 7873 . . . . 5 (𝑛 ∈ ω → Ord 𝑛)
29 ordtr1 6407 . . . . 5 (Ord 𝑛 → ((𝑗𝑖𝑖𝑛) → 𝑗𝑛))
3027, 7, 28, 294syl 19 . . . 4 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → ((𝑗𝑖𝑖𝑛) → 𝑗𝑛))
3123, 24, 30mp2and 698 . . 3 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → 𝑗𝑛)
32 simplr 768 . . 3 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → 𝑖 = suc 𝑗)
3331, 32jca 511 . 2 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → (𝑗𝑛𝑖 = suc 𝑗))
3420, 33bnj1023 34406 1 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗𝑛𝑖 = suc 𝑗))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1534  wex 1774  wcel 2099  wne 2936  wrex 3066  cdif 3942  c0 4319  {csn 4625  Ord word 6363  suc csuc 6366  ωcom 7865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-tr 5261  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-om 7866
This theorem is referenced by:  bnj1110  34608  bnj1128  34616  bnj1145  34619
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