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Theorem bnj1098 31389
 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 1130 . . . . . . 7 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) ↔ (𝑛𝐷𝑖𝑛𝑖 ≠ ∅))
2 df-3an 1113 . . . . . . 7 ((𝑛𝐷𝑖𝑛𝑖 ≠ ∅) ↔ ((𝑛𝐷𝑖𝑛) ∧ 𝑖 ≠ ∅))
31, 2bitri 267 . . . . . 6 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) ↔ ((𝑛𝐷𝑖𝑛) ∧ 𝑖 ≠ ∅))
4 simpr 479 . . . . . . . 8 ((𝑛𝐷𝑖𝑛) → 𝑖𝑛)
5 bnj1098.1 . . . . . . . . . 10 𝐷 = (ω ∖ {∅})
65bnj923 31373 . . . . . . . . 9 (𝑛𝐷𝑛 ∈ ω)
76adantr 474 . . . . . . . 8 ((𝑛𝐷𝑖𝑛) → 𝑛 ∈ ω)
8 elnn 7336 . . . . . . . 8 ((𝑖𝑛𝑛 ∈ ω) → 𝑖 ∈ ω)
94, 7, 8syl2anc 579 . . . . . . 7 ((𝑛𝐷𝑖𝑛) → 𝑖 ∈ ω)
109anim1i 608 . . . . . 6 (((𝑛𝐷𝑖𝑛) ∧ 𝑖 ≠ ∅) → (𝑖 ∈ ω ∧ 𝑖 ≠ ∅))
113, 10sylbi 209 . . . . 5 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑖 ∈ ω ∧ 𝑖 ≠ ∅))
12 nnsuc 7343 . . . . 5 ((𝑖 ∈ ω ∧ 𝑖 ≠ ∅) → ∃𝑗 ∈ ω 𝑖 = suc 𝑗)
1311, 12syl 17 . . . 4 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ∃𝑗 ∈ ω 𝑖 = suc 𝑗)
14 df-rex 3123 . . . . . 6 (∃𝑗 ∈ ω 𝑖 = suc 𝑗 ↔ ∃𝑗(𝑗 ∈ ω ∧ 𝑖 = suc 𝑗))
1514imbi2i 328 . . . . 5 (((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ∃𝑗 ∈ ω 𝑖 = suc 𝑗) ↔ ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ∃𝑗(𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)))
16 19.37v 2096 . . . . 5 (∃𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)) ↔ ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ∃𝑗(𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)))
1715, 16bitr4i 270 . . . 4 (((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ∃𝑗 ∈ ω 𝑖 = suc 𝑗) ↔ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)))
1813, 17mpbi 222 . . 3 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗 ∈ ω ∧ 𝑖 = suc 𝑗))
19 ancr 542 . . 3 (((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)) → ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷))))
2018, 19bnj101 31327 . 2 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → ((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)))
21 vex 3417 . . . . . 6 𝑗 ∈ V
2221bnj216 31336 . . . . 5 (𝑖 = suc 𝑗𝑗𝑖)
2322ad2antlr 718 . . . 4 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → 𝑗𝑖)
24 simpr2 1254 . . . 4 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → 𝑖𝑛)
25 3simpc 1186 . . . . . . 7 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑖𝑛𝑛𝐷))
2625ancomd 455 . . . . . 6 ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑛𝐷𝑖𝑛))
2726adantl 475 . . . . 5 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → (𝑛𝐷𝑖𝑛))
28 nnord 7334 . . . . 5 (𝑛 ∈ ω → Ord 𝑛)
29 ordtr1 6006 . . . . 5 (Ord 𝑛 → ((𝑗𝑖𝑖𝑛) → 𝑗𝑛))
3027, 7, 28, 294syl 19 . . . 4 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → ((𝑗𝑖𝑖𝑛) → 𝑗𝑛))
3123, 24, 30mp2and 690 . . 3 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → 𝑗𝑛)
32 simplr 785 . . 3 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → 𝑖 = suc 𝑗)
3331, 32jca 507 . 2 (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷)) → (𝑗𝑛𝑖 = suc 𝑗))
3420, 33bnj1023 31386 1 𝑗((𝑖 ≠ ∅ ∧ 𝑖𝑛𝑛𝐷) → (𝑗𝑛𝑖 = suc 𝑗))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1111   = wceq 1656  ∃wex 1878   ∈ wcel 2164   ≠ wne 2999  ∃wrex 3118   ∖ cdif 3795  ∅c0 4144  {csn 4397  Ord word 5962  suc csuc 5965  ωcom 7326 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-tr 4976  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-om 7327 This theorem is referenced by:  bnj1110  31585  bnj1128  31593  bnj1145  31596
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