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Theorem bnj168 31995
Description: First-order logic and set theory. Revised to remove dependence on ax-reg 9050. (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 31994 . . . . . . . . 9 (𝑛𝐷 → ∃𝑚 ∈ ω 𝑛 = suc 𝑚)
32anim2i 618 . . . . . . . 8 ((𝑛 ≠ 1o𝑛𝐷) → (𝑛 ≠ 1o ∧ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
4 r19.42v 3350 . . . . . . . 8 (∃𝑚 ∈ ω (𝑛 ≠ 1o𝑛 = suc 𝑚) ↔ (𝑛 ≠ 1o ∧ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
53, 4sylibr 236 . . . . . . 7 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚 ∈ ω (𝑛 ≠ 1o𝑛 = suc 𝑚))
6 neeq1 3078 . . . . . . . . . . 11 (𝑛 = suc 𝑚 → (𝑛 ≠ 1o ↔ suc 𝑚 ≠ 1o))
76biimpac 481 . . . . . . . . . 10 ((𝑛 ≠ 1o𝑛 = suc 𝑚) → suc 𝑚 ≠ 1o)
8 df-1o 8096 . . . . . . . . . . . . 13 1o = suc ∅
98eqeq2i 2834 . . . . . . . . . . . 12 (suc 𝑚 = 1o ↔ suc 𝑚 = suc ∅)
10 nnon 7580 . . . . . . . . . . . . 13 (𝑚 ∈ ω → 𝑚 ∈ On)
11 0elon 6239 . . . . . . . . . . . . 13 ∅ ∈ On
12 suc11 6289 . . . . . . . . . . . . 13 ((𝑚 ∈ On ∧ ∅ ∈ On) → (suc 𝑚 = suc ∅ ↔ 𝑚 = ∅))
1310, 11, 12sylancl 588 . . . . . . . . . . . 12 (𝑚 ∈ ω → (suc 𝑚 = suc ∅ ↔ 𝑚 = ∅))
149, 13syl5rbb 286 . . . . . . . . . . 11 (𝑚 ∈ ω → (𝑚 = ∅ ↔ suc 𝑚 = 1o))
1514necon3bid 3060 . . . . . . . . . 10 (𝑚 ∈ ω → (𝑚 ≠ ∅ ↔ suc 𝑚 ≠ 1o))
167, 15syl5ibr 248 . . . . . . . . 9 (𝑚 ∈ ω → ((𝑛 ≠ 1o𝑛 = suc 𝑚) → 𝑚 ≠ ∅))
1716ancld 553 . . . . . . . 8 (𝑚 ∈ ω → ((𝑛 ≠ 1o𝑛 = suc 𝑚) → ((𝑛 ≠ 1o𝑛 = suc 𝑚) ∧ 𝑚 ≠ ∅)))
1817reximia 3242 . . . . . . 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 471 . . . . . . 7 (((𝑛 ≠ 1o𝑛 = suc 𝑚) ∧ 𝑚 ≠ ∅) ↔ (𝑛 ≠ 1o ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
2120rexbii 3247 . . . . . 6 (∃𝑚 ∈ ω ((𝑛 ≠ 1o𝑛 = suc 𝑚) ∧ 𝑚 ≠ ∅) ↔ ∃𝑚 ∈ ω (𝑛 ≠ 1o ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
2219, 21sylib 220 . . . . 5 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚 ∈ ω (𝑛 ≠ 1o ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
23 simpr 487 . . . . 5 ((𝑛 ≠ 1o ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → (𝑛 = suc 𝑚𝑚 ≠ ∅))
2422, 23bnj31 31984 . . . 4 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚 ∈ ω (𝑛 = suc 𝑚𝑚 ≠ ∅))
25 df-rex 3144 . . . 4 (∃𝑚 ∈ ω (𝑛 = suc 𝑚𝑚 ≠ ∅) ↔ ∃𝑚(𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
2624, 25sylib 220 . . 3 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚(𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
27 simpr 487 . . . . . . 7 ((𝑛 = suc 𝑚𝑚 ≠ ∅) → 𝑚 ≠ ∅)
2827anim2i 618 . . . . . 6 ((𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
291eleq2i 2904 . . . . . . 7 (𝑚𝐷𝑚 ∈ (ω ∖ {∅}))
30 eldifsn 4713 . . . . . . 7 (𝑚 ∈ (ω ∖ {∅}) ↔ (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
3129, 30bitr2i 278 . . . . . 6 ((𝑚 ∈ ω ∧ 𝑚 ≠ ∅) ↔ 𝑚𝐷)
3228, 31sylib 220 . . . . 5 ((𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → 𝑚𝐷)
33 simprl 769 . . . . 5 ((𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → 𝑛 = suc 𝑚)
3432, 33jca 514 . . . 4 ((𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → (𝑚𝐷𝑛 = suc 𝑚))
3534eximi 1831 . . 3 (∃𝑚(𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → ∃𝑚(𝑚𝐷𝑛 = suc 𝑚))
3626, 35syl 17 . 2 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚(𝑚𝐷𝑛 = suc 𝑚))
37 df-rex 3144 . 2 (∃𝑚𝐷 𝑛 = suc 𝑚 ↔ ∃𝑚(𝑚𝐷𝑛 = suc 𝑚))
3836, 37sylibr 236 1 ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚𝐷 𝑛 = suc 𝑚)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wex 1776  wcel 2110  wne 3016  wrex 3139  cdif 3933  c0 4291  {csn 4561  Oncon0 6186  suc csuc 6188  ωcom 7574  1oc1o 8089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-tr 5166  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-om 7575  df-1o 8096
This theorem is referenced by:  bnj600  32186
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