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Theorem bnj168 32717
Description: First-order logic and set theory. Revised to remove dependence on ax-reg 9338. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by NM, 21-Dec-2016.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj168.1 𝐷 = (ω ∖ {∅})
Assertion
Ref Expression
bnj168 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚𝐷 𝑛 = suc 𝑚)
Distinct variable group:   𝑚,𝑛
Allowed substitution hints:   𝐷(𝑚,𝑛)

Proof of Theorem bnj168
StepHypRef Expression
1 bnj168.1 . . . . . . . . . 10 𝐷 = (ω ∖ {∅})
21bnj158 32716 . . . . . . . . 9 (𝑛𝐷 → ∃𝑚 ∈ ω 𝑛 = suc 𝑚)
32anim2i 617 . . . . . . . 8 ((𝑛 ≠ 1o𝑛𝐷) → (𝑛 ≠ 1o ∧ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
4 r19.42v 3277 . . . . . . . 8 (∃𝑚 ∈ ω (𝑛 ≠ 1o𝑛 = suc 𝑚) ↔ (𝑛 ≠ 1o ∧ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
53, 4sylibr 233 . . . . . . 7 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚 ∈ ω (𝑛 ≠ 1o𝑛 = suc 𝑚))
6 neeq1 3006 . . . . . . . . . . 11 (𝑛 = suc 𝑚 → (𝑛 ≠ 1o ↔ suc 𝑚 ≠ 1o))
76biimpac 479 . . . . . . . . . 10 ((𝑛 ≠ 1o𝑛 = suc 𝑚) → suc 𝑚 ≠ 1o)
8 df-1o 8284 . . . . . . . . . . . . 13 1o = suc ∅
98eqeq2i 2751 . . . . . . . . . . . 12 (suc 𝑚 = 1o ↔ suc 𝑚 = suc ∅)
10 nnon 7708 . . . . . . . . . . . . 13 (𝑚 ∈ ω → 𝑚 ∈ On)
11 0elon 6312 . . . . . . . . . . . . 13 ∅ ∈ On
12 suc11 6362 . . . . . . . . . . . . 13 ((𝑚 ∈ On ∧ ∅ ∈ On) → (suc 𝑚 = suc ∅ ↔ 𝑚 = ∅))
1310, 11, 12sylancl 586 . . . . . . . . . . . 12 (𝑚 ∈ ω → (suc 𝑚 = suc ∅ ↔ 𝑚 = ∅))
149, 13bitr2id 284 . . . . . . . . . . 11 (𝑚 ∈ ω → (𝑚 = ∅ ↔ suc 𝑚 = 1o))
1514necon3bid 2988 . . . . . . . . . 10 (𝑚 ∈ ω → (𝑚 ≠ ∅ ↔ suc 𝑚 ≠ 1o))
167, 15syl5ibr 245 . . . . . . . . 9 (𝑚 ∈ ω → ((𝑛 ≠ 1o𝑛 = suc 𝑚) → 𝑚 ≠ ∅))
1716ancld 551 . . . . . . . 8 (𝑚 ∈ ω → ((𝑛 ≠ 1o𝑛 = suc 𝑚) → ((𝑛 ≠ 1o𝑛 = suc 𝑚) ∧ 𝑚 ≠ ∅)))
1817reximia 3174 . . . . . . 7 (∃𝑚 ∈ ω (𝑛 ≠ 1o𝑛 = suc 𝑚) → ∃𝑚 ∈ ω ((𝑛 ≠ 1o𝑛 = suc 𝑚) ∧ 𝑚 ≠ ∅))
195, 18syl 17 . . . . . 6 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚 ∈ ω ((𝑛 ≠ 1o𝑛 = suc 𝑚) ∧ 𝑚 ≠ ∅))
20 anass 469 . . . . . . 7 (((𝑛 ≠ 1o𝑛 = suc 𝑚) ∧ 𝑚 ≠ ∅) ↔ (𝑛 ≠ 1o ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
2120rexbii 3179 . . . . . 6 (∃𝑚 ∈ ω ((𝑛 ≠ 1o𝑛 = suc 𝑚) ∧ 𝑚 ≠ ∅) ↔ ∃𝑚 ∈ ω (𝑛 ≠ 1o ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
2219, 21sylib 217 . . . . 5 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚 ∈ ω (𝑛 ≠ 1o ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
23 simpr 485 . . . . 5 ((𝑛 ≠ 1o ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → (𝑛 = suc 𝑚𝑚 ≠ ∅))
2422, 23bnj31 32706 . . . 4 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚 ∈ ω (𝑛 = suc 𝑚𝑚 ≠ ∅))
25 df-rex 3070 . . . 4 (∃𝑚 ∈ ω (𝑛 = suc 𝑚𝑚 ≠ ∅) ↔ ∃𝑚(𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
2624, 25sylib 217 . . 3 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚(𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
27 simpr 485 . . . . . . 7 ((𝑛 = suc 𝑚𝑚 ≠ ∅) → 𝑚 ≠ ∅)
2827anim2i 617 . . . . . 6 ((𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
291eleq2i 2830 . . . . . . 7 (𝑚𝐷𝑚 ∈ (ω ∖ {∅}))
30 eldifsn 4720 . . . . . . 7 (𝑚 ∈ (ω ∖ {∅}) ↔ (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
3129, 30bitr2i 275 . . . . . 6 ((𝑚 ∈ ω ∧ 𝑚 ≠ ∅) ↔ 𝑚𝐷)
3228, 31sylib 217 . . . . 5 ((𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → 𝑚𝐷)
33 simprl 768 . . . . 5 ((𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → 𝑛 = suc 𝑚)
3432, 33jca 512 . . . 4 ((𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → (𝑚𝐷𝑛 = suc 𝑚))
3534eximi 1837 . . 3 (∃𝑚(𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → ∃𝑚(𝑚𝐷𝑛 = suc 𝑚))
3626, 35syl 17 . 2 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚(𝑚𝐷𝑛 = suc 𝑚))
37 df-rex 3070 . 2 (∃𝑚𝐷 𝑛 = suc 𝑚 ↔ ∃𝑚(𝑚𝐷𝑛 = suc 𝑚))
3836, 37sylibr 233 1 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚𝐷 𝑛 = suc 𝑚)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wex 1782  wcel 2106  wne 2943  wrex 3065  cdif 3883  c0 4256  {csn 4561  Oncon0 6259  suc csuc 6261  ωcom 7702  1oc1o 8277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pr 5350  ax-un 7578
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-pss 3905  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-tr 5191  df-eprel 5490  df-po 5498  df-so 5499  df-fr 5539  df-we 5541  df-ord 6262  df-on 6263  df-lim 6264  df-suc 6265  df-om 7703  df-1o 8284
This theorem is referenced by:  bnj600  32907
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