Theorem bnj1098 30854

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 1049	. . . . . . 7 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) ↔ (𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛 ∧ 𝑖 ≠ ∅))
2		df-3an 1039	. . . . . . 7 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛 ∧ 𝑖 ≠ ∅) ↔ ((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛) ∧ 𝑖 ≠ ∅))
3	1, 2	bitri 264	. . . . . 6 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) ↔ ((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛) ∧ 𝑖 ≠ ∅))
4		simpr 477	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛) → 𝑖 ∈ 𝑛)
5		bnj1098.1	. . . . . . . . . 10 ⊢ 𝐷 = (ω ∖ {∅})
6	5	bnj923 30838	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
7	6	adantr 481	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛) → 𝑛 ∈ ω)
8		elnn 7075	. . . . . . . 8 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝑛 ∈ ω) → 𝑖 ∈ ω)
9	4, 7, 8	syl2anc 693	. . . . . . 7 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛) → 𝑖 ∈ ω)
10	9	anim1i 592	. . . . . 6 ⊢ (((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛) ∧ 𝑖 ≠ ∅) → (𝑖 ∈ ω ∧ 𝑖 ≠ ∅))
11	3, 10	sylbi 207	. . . . 5 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑖 ∈ ω ∧ 𝑖 ≠ ∅))
12		nnsuc 7082	. . . . 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 2918	. . . . . 6 ⊢ (∃𝑗 ∈ ω 𝑖 = suc 𝑗 ↔ ∃𝑗(𝑗 ∈ ω ∧ 𝑖 = suc 𝑗))
15	14	imbi2i 326	. . . . 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 1910	. . . . 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 267	. . . 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 220	. . 3 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑗 ∈ ω ∧ 𝑖 = suc 𝑗))
19		ancr 572	. . 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 30789	. 2 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → ((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)))
21		vex 3203	. . . . . 6 ⊢ 𝑗 ∈ V
22	21	bnj216 30800	. . . . 5 ⊢ (𝑖 = suc 𝑗 → 𝑗 ∈ 𝑖)
23	22	ad2antlr 763	. . . 4 ⊢ (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)) → 𝑗 ∈ 𝑖)
24		simpr2 1068	. . . 4 ⊢ (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)) → 𝑖 ∈ 𝑛)
25		3simpc 1060	. . . . . . 7 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷))
26	25	ancomd 467	. . . . . 6 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛))
27	26	adantl 482	. . . . 5 ⊢ (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)) → (𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛))
28		nnord 7073	. . . . 5 ⊢ (𝑛 ∈ ω → Ord 𝑛)
29		ordtr1 5767	. . . . 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 715	. . 3 ⊢ (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)) → 𝑗 ∈ 𝑛)
32		simplr 792	. . 3 ⊢ (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)) → 𝑖 = suc 𝑗)
33	31, 32	jca 554	. 2 ⊢ (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))
34	20, 33	bnj1023 30851	1 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913   ∖ cdif 3571  ∅c0 3915  {csn 4177  Ord word 5722  suc csuc 5725  ωcom 7065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-om 7066

This theorem is referenced by:  bnj1110  31050  bnj1128  31058  bnj1145  31061