Theorem bnj1098 31389

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 1130	. . . . . . 7 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) ↔ (𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛 ∧ 𝑖 ≠ ∅))
2		df-3an 1113	. . . . . . 7 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛 ∧ 𝑖 ≠ ∅) ↔ ((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛) ∧ 𝑖 ≠ ∅))
3	1, 2	bitri 267	. . . . . 6 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) ↔ ((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛) ∧ 𝑖 ≠ ∅))
4		simpr 479	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛) → 𝑖 ∈ 𝑛)
5		bnj1098.1	. . . . . . . . . 10 ⊢ 𝐷 = (ω ∖ {∅})
6	5	bnj923 31373	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
7	6	adantr 474	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛) → 𝑛 ∈ ω)
8		elnn 7336	. . . . . . . 8 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝑛 ∈ ω) → 𝑖 ∈ ω)
9	4, 7, 8	syl2anc 579	. . . . . . 7 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛) → 𝑖 ∈ ω)
10	9	anim1i 608	. . . . . 6 ⊢ (((𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛) ∧ 𝑖 ≠ ∅) → (𝑖 ∈ ω ∧ 𝑖 ≠ ∅))
11	3, 10	sylbi 209	. . . . 5 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑖 ∈ ω ∧ 𝑖 ≠ ∅))
12		nnsuc 7343	. . . . 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 3123	. . . . . 6 ⊢ (∃𝑗 ∈ ω 𝑖 = suc 𝑗 ↔ ∃𝑗(𝑗 ∈ ω ∧ 𝑖 = suc 𝑗))
15	14	imbi2i 328	. . . . 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 2096	. . . . 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 270	. . . 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 222	. . 3 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑗 ∈ ω ∧ 𝑖 = suc 𝑗))
19		ancr 542	. . 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 31327	. 2 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → ((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)))
21		vex 3417	. . . . . 6 ⊢ 𝑗 ∈ V
22	21	bnj216 31336	. . . . 5 ⊢ (𝑖 = suc 𝑗 → 𝑗 ∈ 𝑖)
23	22	ad2antlr 718	. . . 4 ⊢ (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)) → 𝑗 ∈ 𝑖)
24		simpr2 1254	. . . 4 ⊢ (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)) → 𝑖 ∈ 𝑛)
25		3simpc 1186	. . . . . . 7 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷))
26	25	ancomd 455	. . . . . 6 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛))
27	26	adantl 475	. . . . 5 ⊢ (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)) → (𝑛 ∈ 𝐷 ∧ 𝑖 ∈ 𝑛))
28		nnord 7334	. . . . 5 ⊢ (𝑛 ∈ ω → Ord 𝑛)
29		ordtr1 6006	. . . . 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 690	. . 3 ⊢ (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)) → 𝑗 ∈ 𝑛)
32		simplr 785	. . 3 ⊢ (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)) → 𝑖 = suc 𝑗)
33	31, 32	jca 507	. 2 ⊢ (((𝑗 ∈ ω ∧ 𝑖 = suc 𝑗) ∧ (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))
34	20, 33	bnj1023 31386	1 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))

Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1111   = wceq 1656  ∃wex 1878   ∈ wcel 2164   ≠ wne 2999  ∃wrex 3118   ∖ cdif 3795  ∅c0 4144  {csn 4397  Ord word 5962  suc csuc 5965  ωcom 7326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-tr 4976  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-om 7327

This theorem is referenced by:  bnj1110  31585  bnj1128  31593  bnj1145  31596