Theorem bnj1098 34093

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1098.1	⊢ 𝐷 = (ω ∖ {∅})

Assertion

Ref	Expression
bnj1098	⊢ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))

Distinct variable groups:   𝐷,𝑗   𝑖,𝑗   𝑗,𝑛

Allowed substitution hints:   𝐷(𝑖,𝑛)

Proof of Theorem bnj1098

Step	Hyp	Ref	Expression
1		3anrev 1100	. . . . . . 7 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) ↔ (𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛 ∧ 𝑖 ≠ ∅))
2		df-3an 1088	. . . . . . 7 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛 ∧ 𝑖 ≠ ∅) ↔ ((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛) ∧ 𝑖 ≠ ∅))
3	1, 2	bitri 275	. . . . . 6 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) ↔ ((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛) ∧ 𝑖 ≠ ∅))
4		simpr 484	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛) → 𝑖 ∈ 𝑛)
5		bnj1098.1	. . . . . . . . . 10 ⊢ 𝐷 = (ω ∖ {∅})
6	5	bnj923 34078	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
7	6	adantr 480	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛) → 𝑛 ∈ ω)
8		elnn 7869	. . . . . . . 8 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝑛 ∈ ω) → 𝑖 ∈ ω)
9	4, 7, 8	syl2anc 583	. . . . . . 7 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛) → 𝑖 ∈ ω)
10	9	anim1i 614	. . . . . 6 ⊢ (((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛) ∧ 𝑖 ≠ ∅) → (𝑖 ∈ ω ∧ 𝑖 ≠ ∅))
11	3, 10	sylbi 216	. . . . 5 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑖 ∈ ω ∧ 𝑖 ≠ ∅))
12		nnsuc 7876	. . . . 5 ⊢ ((𝑖 ∈ ω ∧ 𝑖 ≠ ∅) → ∃𝑗 ∈ ω 𝑖 = suc 𝑗)
13	11, 12	syl 17	. . . 4 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → ∃𝑗 ∈ ω 𝑖 = suc 𝑗)
14		df-rex 3070	. . . . . 6 ⊢ (∃𝑗 ∈ ω 𝑖 = suc 𝑗 ↔ ∃𝑗(𝑗 ∈ ω ∧ 𝑖 = suc 𝑗))
15	14	imbi2i 336	. . . . 5 ⊢ (((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → ∃𝑗 ∈ ω 𝑖 = suc 𝑗) ↔ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → ∃𝑗(𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)))
16		19.37v 1994	. . . . 5 ⊢ (∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)) ↔ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → ∃𝑗(𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)))
17	15, 16	bitr4i 278	. . . 4 ⊢ (((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → ∃𝑗 ∈ ω 𝑖 = suc 𝑗) ↔ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)))
18	13, 17	mpbi 229	. . 3 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑗 ∈ ω ∧ 𝑖 = suc 𝑗))
19		ancr 546	. . 3 ⊢ (((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑗 ∈ ω ∧ 𝑖 = suc 𝑗)) → ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → ((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷))))
20	18, 19	bnj101 34033	. 2 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → ((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)))
21		vex 3477	. . . . . 6 ⊢ 𝑗 ∈ V
22	21	bnj216 34042	. . . . 5 ⊢ (𝑖 = suc 𝑗 → 𝑗 ∈ 𝑖)
23	22	ad2antlr 724	. . . 4 ⊢ (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)) → 𝑗 ∈ 𝑖)
24		simpr2 1194	. . . 4 ⊢ (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)) → 𝑖 ∈ 𝑛)
25		3simpc 1149	. . . . . . 7 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷))
26	25	ancomd 461	. . . . . 6 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛))
27	26	adantl 481	. . . . 5 ⊢ (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)) → (𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛))
28		nnord 7866	. . . . 5 ⊢ (𝑛 ∈ ω → Ord 𝑛)
29		ordtr1 6407	. . . . 5 ⊢ (Ord 𝑛 → ((𝑗 ∈ 𝑖 ∧ 𝑖 ∈ 𝑛) → 𝑗 ∈ 𝑛))
30	27, 7, 28, 29	4syl 19	. . . 4 ⊢ (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)) → ((𝑗 ∈ 𝑖 ∧ 𝑖 ∈ 𝑛) → 𝑗 ∈ 𝑛))
31	23, 24, 30	mp2and 696	. . 3 ⊢ (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)) → 𝑗 ∈ 𝑛)
32		simplr 766	. . 3 ⊢ (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)) → 𝑖 = suc 𝑗)
33	31, 32	jca 511	. 2 ⊢ (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))
34	20, 33	bnj1023 34090	1 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1780   ∈ wcel 2105   ≠ wne 2939  ∃wrex 3069   ∖ cdif 3945  ∅c0 4322  {csn 4628  Ord word 6363  suc csuc 6366  ωcom 7858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7728

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-om 7859

This theorem is referenced by:  bnj1110  34292  bnj1128  34300  bnj1145  34303