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Theorem bnj168 30772
Description: First-order logic and set theory. Revised to remove dependence on ax-reg 8482. (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 30771 . . . . . . . . 9 (𝑛𝐷 → ∃𝑚 ∈ ω 𝑛 = suc 𝑚)
32anim2i 592 . . . . . . . 8 ((𝑛 ≠ 1𝑜𝑛𝐷) → (𝑛 ≠ 1𝑜 ∧ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
4 r19.42v 3087 . . . . . . . 8 (∃𝑚 ∈ ω (𝑛 ≠ 1𝑜𝑛 = suc 𝑚) ↔ (𝑛 ≠ 1𝑜 ∧ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
53, 4sylibr 224 . . . . . . 7 ((𝑛 ≠ 1𝑜𝑛𝐷) → ∃𝑚 ∈ ω (𝑛 ≠ 1𝑜𝑛 = suc 𝑚))
6 neeq1 2853 . . . . . . . . . . 11 (𝑛 = suc 𝑚 → (𝑛 ≠ 1𝑜 ↔ suc 𝑚 ≠ 1𝑜))
76biimpac 503 . . . . . . . . . 10 ((𝑛 ≠ 1𝑜𝑛 = suc 𝑚) → suc 𝑚 ≠ 1𝑜)
8 df-1o 7545 . . . . . . . . . . . . 13 1𝑜 = suc ∅
98eqeq2i 2632 . . . . . . . . . . . 12 (suc 𝑚 = 1𝑜 ↔ suc 𝑚 = suc ∅)
10 nnon 7056 . . . . . . . . . . . . 13 (𝑚 ∈ ω → 𝑚 ∈ On)
11 0elon 5766 . . . . . . . . . . . . 13 ∅ ∈ On
12 suc11 5819 . . . . . . . . . . . . 13 ((𝑚 ∈ On ∧ ∅ ∈ On) → (suc 𝑚 = suc ∅ ↔ 𝑚 = ∅))
1310, 11, 12sylancl 693 . . . . . . . . . . . 12 (𝑚 ∈ ω → (suc 𝑚 = suc ∅ ↔ 𝑚 = ∅))
149, 13syl5rbb 273 . . . . . . . . . . 11 (𝑚 ∈ ω → (𝑚 = ∅ ↔ suc 𝑚 = 1𝑜))
1514necon3bid 2835 . . . . . . . . . 10 (𝑚 ∈ ω → (𝑚 ≠ ∅ ↔ suc 𝑚 ≠ 1𝑜))
167, 15syl5ibr 236 . . . . . . . . 9 (𝑚 ∈ ω → ((𝑛 ≠ 1𝑜𝑛 = suc 𝑚) → 𝑚 ≠ ∅))
1716ancld 575 . . . . . . . 8 (𝑚 ∈ ω → ((𝑛 ≠ 1𝑜𝑛 = suc 𝑚) → ((𝑛 ≠ 1𝑜𝑛 = suc 𝑚) ∧ 𝑚 ≠ ∅)))
1817reximia 3006 . . . . . . 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 680 . . . . . . 7 (((𝑛 ≠ 1𝑜𝑛 = suc 𝑚) ∧ 𝑚 ≠ ∅) ↔ (𝑛 ≠ 1𝑜 ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
2120rexbii 3037 . . . . . 6 (∃𝑚 ∈ ω ((𝑛 ≠ 1𝑜𝑛 = suc 𝑚) ∧ 𝑚 ≠ ∅) ↔ ∃𝑚 ∈ ω (𝑛 ≠ 1𝑜 ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
2219, 21sylib 208 . . . . 5 ((𝑛 ≠ 1𝑜𝑛𝐷) → ∃𝑚 ∈ ω (𝑛 ≠ 1𝑜 ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
23 simpr 477 . . . . 5 ((𝑛 ≠ 1𝑜 ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → (𝑛 = suc 𝑚𝑚 ≠ ∅))
2422, 23bnj31 30759 . . . 4 ((𝑛 ≠ 1𝑜𝑛𝐷) → ∃𝑚 ∈ ω (𝑛 = suc 𝑚𝑚 ≠ ∅))
25 df-rex 2915 . . . 4 (∃𝑚 ∈ ω (𝑛 = suc 𝑚𝑚 ≠ ∅) ↔ ∃𝑚(𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
2624, 25sylib 208 . . 3 ((𝑛 ≠ 1𝑜𝑛𝐷) → ∃𝑚(𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)))
27 simpr 477 . . . . . . 7 ((𝑛 = suc 𝑚𝑚 ≠ ∅) → 𝑚 ≠ ∅)
2827anim2i 592 . . . . . 6 ((𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
291eleq2i 2691 . . . . . . 7 (𝑚𝐷𝑚 ∈ (ω ∖ {∅}))
30 eldifsn 4308 . . . . . . 7 (𝑚 ∈ (ω ∖ {∅}) ↔ (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
3129, 30bitr2i 265 . . . . . 6 ((𝑚 ∈ ω ∧ 𝑚 ≠ ∅) ↔ 𝑚𝐷)
3228, 31sylib 208 . . . . 5 ((𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → 𝑚𝐷)
33 simprl 793 . . . . 5 ((𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → 𝑛 = suc 𝑚)
3432, 33jca 554 . . . 4 ((𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → (𝑚𝐷𝑛 = suc 𝑚))
3534eximi 1760 . . 3 (∃𝑚(𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → ∃𝑚(𝑚𝐷𝑛 = suc 𝑚))
3626, 35syl 17 . 2 ((𝑛 ≠ 1𝑜𝑛𝐷) → ∃𝑚(𝑚𝐷𝑛 = suc 𝑚))
37 df-rex 2915 . 2 (∃𝑚𝐷 𝑛 = suc 𝑚 ↔ ∃𝑚(𝑚𝐷𝑛 = suc 𝑚))
3836, 37sylibr 224 1 ((𝑛 ≠ 1𝑜𝑛𝐷) → ∃𝑚𝐷 𝑛 = suc 𝑚)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wex 1702  wcel 1988  wne 2791  wrex 2910  cdif 3564  c0 3907  {csn 4168  Oncon0 5711  suc csuc 5713  ωcom 7050  1𝑜c1o 7538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-tr 4744  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-om 7051  df-1o 7545
This theorem is referenced by:  bnj600  30963
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