Theorem bnj970 34961

Description: Technical lemma for bnj69 35024. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj970.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj970.10	⊢ 𝐷 = (ω ∖ {∅})

Assertion

Ref	Expression
bnj970	⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝑝 ∈ 𝐷)

Proof of Theorem bnj970

Step	Hyp	Ref	Expression
1		bnj970.3	. . . . 5 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
2	1	bnj1232 34817	. . . 4 ⊢ (𝜒 → 𝑛 ∈ 𝐷)
3	2	3ad2ant1 1134	. . 3 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝑛 ∈ 𝐷)
4	3	adantl 481	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝑛 ∈ 𝐷)
5		simpr3 1197	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝑝 = suc 𝑛)
6		bnj970.10	. . . . 5 ⊢ 𝐷 = (ω ∖ {∅})
7	6	bnj923 34782	. . . 4 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
8		peano2 7912	. . . . 5 ⊢ (𝑛 ∈ ω → suc 𝑛 ∈ ω)
9		eleq1 2829	. . . . 5 ⊢ (𝑝 = suc 𝑛 → (𝑝 ∈ ω ↔ suc 𝑛 ∈ ω))
10		bianir 1059	. . . . 5 ⊢ ((suc 𝑛 ∈ ω ∧ (𝑝 ∈ ω ↔ suc 𝑛 ∈ ω)) → 𝑝 ∈ ω)
11	8, 9, 10	syl2an 596	. . . 4 ⊢ ((𝑛 ∈ ω ∧ 𝑝 = suc 𝑛) → 𝑝 ∈ ω)
12	7, 11	sylan 580	. . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛) → 𝑝 ∈ ω)
13		df-suc 6390	. . . . . 6 ⊢ suc 𝑛 = (𝑛 ∪ {𝑛})
14	13	eqeq2i 2750	. . . . 5 ⊢ (𝑝 = suc 𝑛 ↔ 𝑝 = (𝑛 ∪ {𝑛}))
15		ssun2 4179	. . . . . . 7 ⊢ {𝑛} ⊆ (𝑛 ∪ {𝑛})
16		vex 3484	. . . . . . . 8 ⊢ 𝑛 ∈ V
17	16	snnz 4776	. . . . . . 7 ⊢ {𝑛} ≠ ∅
18		ssn0 4404	. . . . . . 7 ⊢ (({𝑛} ⊆ (𝑛 ∪ {𝑛}) ∧ {𝑛} ≠ ∅) → (𝑛 ∪ {𝑛}) ≠ ∅)
19	15, 17, 18	mp2an 692	. . . . . 6 ⊢ (𝑛 ∪ {𝑛}) ≠ ∅
20		neeq1 3003	. . . . . 6 ⊢ (𝑝 = (𝑛 ∪ {𝑛}) → (𝑝 ≠ ∅ ↔ (𝑛 ∪ {𝑛}) ≠ ∅))
21	19, 20	mpbiri 258	. . . . 5 ⊢ (𝑝 = (𝑛 ∪ {𝑛}) → 𝑝 ≠ ∅)
22	14, 21	sylbi 217	. . . 4 ⊢ (𝑝 = suc 𝑛 → 𝑝 ≠ ∅)
23	22	adantl 481	. . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛) → 𝑝 ≠ ∅)
24	6	eleq2i 2833	. . . 4 ⊢ (𝑝 ∈ 𝐷 ↔ 𝑝 ∈ (ω ∖ {∅}))
25		eldifsn 4786	. . . 4 ⊢ (𝑝 ∈ (ω ∖ {∅}) ↔ (𝑝 ∈ ω ∧ 𝑝 ≠ ∅))
26	24, 25	bitri 275	. . 3 ⊢ (𝑝 ∈ 𝐷 ↔ (𝑝 ∈ ω ∧ 𝑝 ≠ ∅))
27	12, 23, 26	sylanbrc 583	. 2 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛) → 𝑝 ∈ 𝐷)
28	4, 5, 27	syl2anc 584	1 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝑝 ∈ 𝐷)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940   ∖ cdif 3948   ∪ cun 3949   ⊆ wss 3951  ∅c0 4333  {csn 4626  suc csuc 6386   Fn wfn 6556  ωcom 7887   ∧ w-bnj17 34700   FrSe w-bnj15 34706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-om 7888  df-bnj17 34701

This theorem is referenced by:  bnj910  34962