Theorem bnj970 35103

Description: Technical lemma for bnj69 35166. 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 34959	. . . 4 ⊢ (𝜒 → 𝑛 ∈ 𝐷)
3	2	3ad2ant1 1133	. . 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 34924	. . . 4 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
8		peano2 7832	. . . . 5 ⊢ (𝑛 ∈ ω → suc 𝑛 ∈ ω)
9		eleq1 2824	. . . . 5 ⊢ (𝑝 = suc 𝑛 → (𝑝 ∈ ω ↔ suc 𝑛 ∈ ω))
10		bianir 1058	. . . . 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 6323	. . . . . 6 ⊢ suc 𝑛 = (𝑛 ∪ {𝑛})
14	13	eqeq2i 2749	. . . . 5 ⊢ (𝑝 = suc 𝑛 ↔ 𝑝 = (𝑛 ∪ {𝑛}))
15		ssun2 4131	. . . . . . 7 ⊢ {𝑛} ⊆ (𝑛 ∪ {𝑛})
16		vex 3444	. . . . . . . 8 ⊢ 𝑛 ∈ V
17	16	snnz 4733	. . . . . . 7 ⊢ {𝑛} ≠ ∅
18		ssn0 4356	. . . . . . 7 ⊢ (({𝑛} ⊆ (𝑛 ∪ {𝑛}) ∧ {𝑛} ≠ ∅) → (𝑛 ∪ {𝑛}) ≠ ∅)
19	15, 17, 18	mp2an 692	. . . . . 6 ⊢ (𝑛 ∪ {𝑛}) ≠ ∅
20		neeq1 2994	. . . . . 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 2828	. . . 4 ⊢ (𝑝 ∈ 𝐷 ↔ 𝑝 ∈ (ω ∖ {∅}))
25		eldifsn 4742	. . . 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 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2932   ∖ cdif 3898   ∪ cun 3899   ⊆ wss 3901  ∅c0 4285  {csn 4580  suc csuc 6319   Fn wfn 6487  ωcom 7808   ∧ w-bnj17 34842   FrSe w-bnj15 34848

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-om 7809  df-bnj17 34843

This theorem is referenced by:  bnj910  35104