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Theorem bnj1098 30615
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 1047 . . . . . . 7 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) ↔ (𝑛𝐷𝑖𝑛𝑖 ≠ ∅))
2 df-3an 1038 . . . . . . 7 ((𝑛𝐷𝑖𝑛𝑖 ≠ ∅) ↔ ((𝑛𝐷𝑖𝑛) ∧ 𝑖 ≠ ∅))
31, 2bitri 264 . . . . . 6 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) ↔ ((𝑛𝐷𝑖𝑛) ∧ 𝑖 ≠ ∅))
4 simpr 477 . . . . . . . 8 ((𝑛𝐷𝑖𝑛) → 𝑖𝑛)
5 bnj1098.1 . . . . . . . . . 10 𝐷 = (ω ∖ {∅})
65bnj923 30599 . . . . . . . . 9 (𝑛𝐷𝑛 ∈ ω)
76adantr 481 . . . . . . . 8 ((𝑛𝐷𝑖𝑛) → 𝑛 ∈ ω)
8 elnn 7037 . . . . . . . 8 ((𝑖𝑛𝑛 ∈ ω) → 𝑖 ∈ ω)
94, 7, 8syl2anc 692 . . . . . . 7 ((𝑛𝐷𝑖𝑛) → 𝑖 ∈ ω)
109anim1i 591 . . . . . 6 (((𝑛𝐷𝑖𝑛) ∧ 𝑖 ≠ ∅) → (𝑖 ∈ ω ∧ 𝑖 ≠ ∅))
113, 10sylbi 207 . . . . 5 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑖 ∈ ω ∧ 𝑖 ≠ ∅))
12 nnsuc 7044 . . . . 5 ((𝑖 ∈ ω ∧ 𝑖 ≠ ∅) → ∃𝑗 ∈ ω 𝑖 = suc 𝑗)
1311, 12syl 17 . . . 4 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ∃𝑗 ∈ ω 𝑖 = suc 𝑗)
14 df-rex 2914 . . . . . 6 (∃𝑗 ∈ ω 𝑖 = suc 𝑗 ↔ ∃𝑗(𝑗 ∈ ω ∧ 𝑖 = suc 𝑗))
1514imbi2i 326 . . . . 5 (((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ∃𝑗 ∈ ω 𝑖 = suc 𝑗) ↔ ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ∃𝑗(𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)))
16 19.37v 1907 . . . . 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 30550 . 2 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)))
21 vex 3193 . . . . . 6 𝑗 ∈ V
2221bnj216 30561 . . . . 5 (𝑖 = suc 𝑗𝑗𝑖)
2322ad2antlr 762 . . . 4 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → 𝑗𝑖)
24 simpr2 1066 . . . 4 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → 𝑖𝑛)
25 3simpc 1058 . . . . . . 7 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑖𝑛𝑛𝐷))
2625ancomd 467 . . . . . 6 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑛𝐷𝑖𝑛))
2726adantl 482 . . . . 5 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → (𝑛𝐷𝑖𝑛))
28 nnord 7035 . . . . 5 (𝑛 ∈ ω → Ord 𝑛)
29 ordtr1 5736 . . . . 5 (Ord 𝑛 → ((𝑗𝑖𝑖𝑛) → 𝑗𝑛))
3027, 7, 28, 294syl 19 . . . 4 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → ((𝑗𝑖𝑖𝑛) → 𝑗𝑛))
3123, 24, 30mp2and 714 . . 3 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → 𝑗𝑛)
32 simplr 791 . . 3 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → 𝑖 = suc 𝑗)
3331, 32jca 554 . 2 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → (𝑗𝑛𝑖 = suc 𝑗))
3420, 33bnj1023 30612 1 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗𝑛𝑖 = suc 𝑗))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wne 2790  wrex 2909  cdif 3557  c0 3897  {csn 4155  Ord word 5691  suc csuc 5694  ωcom 7027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-tr 4723  df-eprel 4995  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-om 7028
This theorem is referenced by:  bnj1110  30811  bnj1128  30819  bnj1145  30822
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