Theorem bnj970 35144

Description: Technical lemma for bnj69 35207. 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 35000	. . . 4 ⊢ (𝜒 → 𝑛 ∈ 𝐷)
3	2	3ad2ant1 1140	. . 3 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝑛 ∈ 𝐷)
4	3	adantl 483	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝑛 ∈ 𝐷)
5		simpr3 1204	. 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 34966	. . . 4 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
8		peano2 7834	. . . . 5 ⊢ (𝑛 ∈ ω → suc 𝑛 ∈ ω)
9		eleq1 2829	. . . . 5 ⊢ (𝑝 = suc 𝑛 → (𝑝 ∈ ω ↔ suc 𝑛 ∈ ω))
10		bianir 1065	. . . . 5 ⊢ ((suc 𝑛 ∈ ω ∧ (𝑝 ∈ ω ↔ suc 𝑛 ∈ ω)) → 𝑝 ∈ ω)
11	8, 9, 10	syl2an 603	. . . 4 ⊢ ((𝑛 ∈ ω ∧ 𝑝 = suc 𝑛) → 𝑝 ∈ ω)
12	7, 11	sylan 587	. . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛) → 𝑝 ∈ ω)
13		df-suc 6320	. . . . . 6 ⊢ suc 𝑛 = (𝑛 ∪ {𝑛})
14	13	eqeq2i 2754	. . . . 5 ⊢ (𝑝 = suc 𝑛 ↔ 𝑝 = (𝑛 ∪ {𝑛}))
15		ssun2 4111	. . . . . . 7 ⊢ {𝑛} ⊆ (𝑛 ∪ {𝑛})
16		vex 3437	. . . . . . . 8 ⊢ 𝑛 ∈ V
17	16	snnz 4711	. . . . . . 7 ⊢ {𝑛} ≠ ∅
18		ssn0 4335	. . . . . . 7 ⊢ (({𝑛} ⊆ (𝑛 ∪ {𝑛}) ∧ {𝑛} ≠ ∅) → (𝑛 ∪ {𝑛}) ≠ ∅)
19	15, 17, 18	mp2an 699	. . . . . 6 ⊢ (𝑛 ∪ {𝑛}) ≠ ∅
20		neeq1 2998	. . . . . 6 ⊢ (𝑝 = (𝑛 ∪ {𝑛}) → (𝑝 ≠ ∅ ↔ (𝑛 ∪ {𝑛}) ≠ ∅))
21	19, 20	mpbiri 260	. . . . 5 ⊢ (𝑝 = (𝑛 ∪ {𝑛}) → 𝑝 ≠ ∅)
22	14, 21	sylbi 219	. . . 4 ⊢ (𝑝 = suc 𝑛 → 𝑝 ≠ ∅)
23	22	adantl 483	. . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛) → 𝑝 ≠ ∅)
24	6	eleq2i 2833	. . . 4 ⊢ (𝑝 ∈ 𝐷 ↔ 𝑝 ∈ (ω ∖ {∅}))
25		eldifsn 4722	. . . 4 ⊢ (𝑝 ∈ (ω ∖ {∅}) ↔ (𝑝 ∈ ω ∧ 𝑝 ≠ ∅))
26	24, 25	bitri 277	. . 3 ⊢ (𝑝 ∈ 𝐷 ↔ (𝑝 ∈ ω ∧ 𝑝 ≠ ∅))
27	12, 23, 26	sylanbrc 590	. 2 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛) → 𝑝 ∈ 𝐷)
28	4, 5, 27	syl2anc 591	1 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝑝 ∈ 𝐷)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121   ≠ wne 2936   ∖ cdif 3882   ∪ cun 3883   ⊆ wss 3885  ∅c0 4264  {csn 4558  suc csuc 6316   Fn wfn 6484  ωcom 7810   ∧ w-bnj17 34884   FrSe w-bnj15 34890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-tr 5183  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-om 7811  df-bnj17 34885

This theorem is referenced by:  bnj910  35145