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Theorem bnj168 30052
 Description: First-order logic and set theory. Revised to remove dependence on ax-reg 8380. (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 30051 . . . . . . . . 9 (𝑛𝐷 → ∃𝑚 ∈ ω 𝑛 = suc 𝑚)
32anim2i 591 . . . . . . . 8 ((𝑛 ≠ 1𝑜𝑛𝐷) → (𝑛 ≠ 1𝑜 ∧ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
4 r19.42v 3073 . . . . . . . 8 (∃𝑚 ∈ ω (𝑛 ≠ 1𝑜𝑛 = suc 𝑚) ↔ (𝑛 ≠ 1𝑜 ∧ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
53, 4sylibr 223 . . . . . . 7 ((𝑛 ≠ 1𝑜𝑛𝐷) → ∃𝑚 ∈ ω (𝑛 ≠ 1𝑜𝑛 = suc 𝑚))
6 neeq1 2844 . . . . . . . . . . 11 (𝑛 = suc 𝑚 → (𝑛 ≠ 1𝑜 ↔ suc 𝑚 ≠ 1𝑜))
76biimpac 502 . . . . . . . . . 10 ((𝑛 ≠ 1𝑜𝑛 = suc 𝑚) → suc 𝑚 ≠ 1𝑜)
8 df-1o 7447 . . . . . . . . . . . . 13 1𝑜 = suc ∅
98eqeq2i 2622 . . . . . . . . . . . 12 (suc 𝑚 = 1𝑜 ↔ suc 𝑚 = suc ∅)
10 nnon 6963 . . . . . . . . . . . . 13 (𝑚 ∈ ω → 𝑚 ∈ On)
11 0elon 5695 . . . . . . . . . . . . 13 ∅ ∈ On
12 suc11 5748 . . . . . . . . . . . . 13 ((𝑚 ∈ On ∧ ∅ ∈ On) → (suc 𝑚 = suc ∅ ↔ 𝑚 = ∅))
1310, 11, 12sylancl 693 . . . . . . . . . . . 12 (𝑚 ∈ ω → (suc 𝑚 = suc ∅ ↔ 𝑚 = ∅))
149, 13syl5rbb 272 . . . . . . . . . . 11 (𝑚 ∈ ω → (𝑚 = ∅ ↔ suc 𝑚 = 1𝑜))
1514necon3bid 2826 . . . . . . . . . 10 (𝑚 ∈ ω → (𝑚 ≠ ∅ ↔ suc 𝑚 ≠ 1𝑜))
167, 15syl5ibr 235 . . . . . . . . 9 (𝑚 ∈ ω → ((𝑛 ≠ 1𝑜𝑛 = suc 𝑚) → 𝑚 ≠ ∅))
1716ancld 574 . . . . . . . 8 (𝑚 ∈ ω → ((𝑛 ≠ 1𝑜𝑛 = suc 𝑚) → ((𝑛 ≠ 1𝑜𝑛 = suc 𝑚) ∧ 𝑚 ≠ ∅)))
1817reximia 2992 . . . . . . 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 679 . . . . . . 7 (((𝑛 ≠ 1𝑜𝑛 = suc 𝑚) ∧ 𝑚 ≠ ∅) ↔ (𝑛 ≠ 1𝑜 ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
2120rexbii 3023 . . . . . 6 (∃𝑚 ∈ ω ((𝑛 ≠ 1𝑜𝑛 = suc 𝑚) ∧ 𝑚 ≠ ∅) ↔ ∃𝑚 ∈ ω (𝑛 ≠ 1𝑜 ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
2219, 21sylib 207 . . . . 5 ((𝑛 ≠ 1𝑜𝑛𝐷) → ∃𝑚 ∈ ω (𝑛 ≠ 1𝑜 ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
23 simpr 476 . . . . 5 ((𝑛 ≠ 1𝑜 ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → (𝑛 = suc 𝑚𝑚 ≠ ∅))
2422, 23bnj31 30039 . . . 4 ((𝑛 ≠ 1𝑜𝑛𝐷) → ∃𝑚 ∈ ω (𝑛 = suc 𝑚𝑚 ≠ ∅))
25 df-rex 2902 . . . 4 (∃𝑚 ∈ ω (𝑛 = suc 𝑚𝑚 ≠ ∅) ↔ ∃𝑚(𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
2624, 25sylib 207 . . 3 ((𝑛 ≠ 1𝑜𝑛𝐷) → ∃𝑚(𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
27 simpr 476 . . . . . . 7 ((𝑛 = suc 𝑚𝑚 ≠ ∅) → 𝑚 ≠ ∅)
2827anim2i 591 . . . . . 6 ((𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
291eleq2i 2680 . . . . . . 7 (𝑚𝐷𝑚 ∈ (ω ∖ {∅}))
30 eldifsn 4260 . . . . . . 7 (𝑚 ∈ (ω ∖ {∅}) ↔ (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
3129, 30bitr2i 264 . . . . . 6 ((𝑚 ∈ ω ∧ 𝑚 ≠ ∅) ↔ 𝑚𝐷)
3228, 31sylib 207 . . . . 5 ((𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → 𝑚𝐷)
33 simprl 790 . . . . 5 ((𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → 𝑛 = suc 𝑚)
3432, 33jca 553 . . . 4 ((𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → (𝑚𝐷𝑛 = suc 𝑚))
3534eximi 1752 . . 3 (∃𝑚(𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → ∃𝑚(𝑚𝐷𝑛 = suc 𝑚))
3626, 35syl 17 . 2 ((𝑛 ≠ 1𝑜𝑛𝐷) → ∃𝑚(𝑚𝐷𝑛 = suc 𝑚))
37 df-rex 2902 . 2 (∃𝑚𝐷 𝑛 = suc 𝑚 ↔ ∃𝑚(𝑚𝐷𝑛 = suc 𝑚))
3836, 37sylibr 223 1 ((𝑛 ≠ 1𝑜𝑛𝐷) → ∃𝑚𝐷 𝑛 = suc 𝑚)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   ∖ cdif 3537  ∅c0 3874  {csn 4125  Oncon0 5640  suc csuc 5642  ωcom 6957  1𝑜c1o 7440 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-om 6958  df-1o 7447 This theorem is referenced by:  bnj600  30243
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