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Theorem bnj1098 30980
 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 1067 . . . . . . 7 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) ↔ (𝑛𝐷𝑖𝑛𝑖 ≠ ∅))
2 df-3an 1056 . . . . . . 7 ((𝑛𝐷𝑖𝑛𝑖 ≠ ∅) ↔ ((𝑛𝐷𝑖𝑛) ∧ 𝑖 ≠ ∅))
31, 2bitri 264 . . . . . 6 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) ↔ ((𝑛𝐷𝑖𝑛) ∧ 𝑖 ≠ ∅))
4 simpr 476 . . . . . . . 8 ((𝑛𝐷𝑖𝑛) → 𝑖𝑛)
5 bnj1098.1 . . . . . . . . . 10 𝐷 = (ω ∖ {∅})
65bnj923 30964 . . . . . . . . 9 (𝑛𝐷𝑛 ∈ ω)
76adantr 480 . . . . . . . 8 ((𝑛𝐷𝑖𝑛) → 𝑛 ∈ ω)
8 elnn 7117 . . . . . . . 8 ((𝑖𝑛𝑛 ∈ ω) → 𝑖 ∈ ω)
94, 7, 8syl2anc 694 . . . . . . 7 ((𝑛𝐷𝑖𝑛) → 𝑖 ∈ ω)
109anim1i 591 . . . . . 6 (((𝑛𝐷𝑖𝑛) ∧ 𝑖 ≠ ∅) → (𝑖 ∈ ω ∧ 𝑖 ≠ ∅))
113, 10sylbi 207 . . . . 5 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑖 ∈ ω ∧ 𝑖 ≠ ∅))
12 nnsuc 7124 . . . . 5 ((𝑖 ∈ ω ∧ 𝑖 ≠ ∅) → ∃𝑗 ∈ ω 𝑖 = suc 𝑗)
1311, 12syl 17 . . . 4 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ∃𝑗 ∈ ω 𝑖 = suc 𝑗)
14 df-rex 2947 . . . . . 6 (∃𝑗 ∈ ω 𝑖 = suc 𝑗 ↔ ∃𝑗(𝑗 ∈ ω ∧ 𝑖 = suc 𝑗))
1514imbi2i 325 . . . . 5 (((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ∃𝑗 ∈ ω 𝑖 = suc 𝑗) ↔ ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ∃𝑗(𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)))
16 19.37v 1966 . . . . 5 (∃𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)) ↔ ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ∃𝑗(𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)))
1715, 16bitr4i 267 . . . 4 (((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ∃𝑗 ∈ ω 𝑖 = suc 𝑗) ↔ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)))
1813, 17mpbi 220 . . 3 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗 ∈ ω ∧ 𝑖 = suc 𝑗))
19 ancr 571 . . 3 (((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)) → ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷))))
2018, 19bnj101 30917 . 2 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)))
21 vex 3234 . . . . . 6 𝑗 ∈ V
2221bnj216 30926 . . . . 5 (𝑖 = suc 𝑗𝑗𝑖)
2322ad2antlr 763 . . . 4 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → 𝑗𝑖)
24 simpr2 1088 . . . 4 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → 𝑖𝑛)
25 3simpc 1080 . . . . . . 7 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑖𝑛𝑛𝐷))
2625ancomd 466 . . . . . 6 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑛𝐷𝑖𝑛))
2726adantl 481 . . . . 5 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → (𝑛𝐷𝑖𝑛))
28 nnord 7115 . . . . 5 (𝑛 ∈ ω → Ord 𝑛)
29 ordtr1 5805 . . . . 5 (Ord 𝑛 → ((𝑗𝑖𝑖𝑛) → 𝑗𝑛))
3027, 7, 28, 294syl 19 . . . 4 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → ((𝑗𝑖𝑖𝑛) → 𝑗𝑛))
3123, 24, 30mp2and 715 . . 3 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → 𝑗𝑛)
32 simplr 807 . . 3 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → 𝑖 = suc 𝑗)
3331, 32jca 553 . 2 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → (𝑗𝑛𝑖 = suc 𝑗))
3420, 33bnj1023 30977 1 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗𝑛𝑖 = suc 𝑗))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523  ∃wex 1744   ∈ wcel 2030   ≠ wne 2823  ∃wrex 2942   ∖ cdif 3604  ∅c0 3948  {csn 4210  Ord word 5760  suc csuc 5763  ωcom 7107 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-om 7108 This theorem is referenced by:  bnj1110  31176  bnj1128  31184  bnj1145  31187
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