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Theorem bnj168 30541
Description: First-order logic and set theory. Revised to remove dependence on ax-reg 8449. (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 30540 . . . . . . . . 9 (𝑛𝐷 → ∃𝑚 ∈ ω 𝑛 = suc 𝑚)
32anim2i 592 . . . . . . . 8 ((𝑛 ≠ 1𝑜𝑛𝐷) → (𝑛 ≠ 1𝑜 ∧ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
4 r19.42v 3085 . . . . . . . 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 2852 . . . . . . . . . . 11 (𝑛 = suc 𝑚 → (𝑛 ≠ 1𝑜 ↔ suc 𝑚 ≠ 1𝑜))
76biimpac 503 . . . . . . . . . 10 ((𝑛 ≠ 1𝑜𝑛 = suc 𝑚) → suc 𝑚 ≠ 1𝑜)
8 df-1o 7512 . . . . . . . . . . . . 13 1𝑜 = suc ∅
98eqeq2i 2633 . . . . . . . . . . . 12 (suc 𝑚 = 1𝑜 ↔ suc 𝑚 = suc ∅)
10 nnon 7025 . . . . . . . . . . . . 13 (𝑚 ∈ ω → 𝑚 ∈ On)
11 0elon 5742 . . . . . . . . . . . . 13 ∅ ∈ On
12 suc11 5795 . . . . . . . . . . . . 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 2834 . . . . . . . . . 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 3004 . . . . . . 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 3035 . . . . . 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 30528 . . . 4 ((𝑛 ≠ 1𝑜𝑛𝐷) → ∃𝑚 ∈ ω (𝑛 = suc 𝑚𝑚 ≠ ∅))
25 df-rex 2913 . . . 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 2690 . . . . . . 7 (𝑚𝐷𝑚 ∈ (ω ∖ {∅}))
30 eldifsn 4292 . . . . . . 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 1759 . . 3 (∃𝑚(𝑚 ∈ ω ∧ (𝑛 = suc 𝑚𝑚 ≠ ∅)) → ∃𝑚(𝑚𝐷𝑛 = suc 𝑚))
3626, 35syl 17 . 2 ((𝑛 ≠ 1𝑜𝑛𝐷) → ∃𝑚(𝑚𝐷𝑛 = suc 𝑚))
37 df-rex 2913 . 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 1480  wex 1701  wcel 1987  wne 2790  wrex 2908  cdif 3556  c0 3896  {csn 4153  Oncon0 5687  suc csuc 5689  ωcom 7019  1𝑜c1o 7505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-om 7020  df-1o 7512
This theorem is referenced by:  bnj600  30732
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