Theorem bnj970 35105

Description: Technical lemma for bnj69 35168. 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 34961	. . . 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 1198	. 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 34926	. . . 4 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
8		peano2 7834	. . . . 5 ⊢ (𝑛 ∈ ω → suc 𝑛 ∈ ω)
9		eleq1 2825	. . . . 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 597	. . . 4 ⊢ ((𝑛 ∈ ω ∧ 𝑝 = suc 𝑛) → 𝑝 ∈ ω)
12	7, 11	sylan 581	. . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛) → 𝑝 ∈ ω)
13		df-suc 6324	. . . . . 6 ⊢ suc 𝑛 = (𝑛 ∪ {𝑛})
14	13	eqeq2i 2750	. . . . 5 ⊢ (𝑝 = suc 𝑛 ↔ 𝑝 = (𝑛 ∪ {𝑛}))
15		ssun2 4132	. . . . . . 7 ⊢ {𝑛} ⊆ (𝑛 ∪ {𝑛})
16		vex 3445	. . . . . . . 8 ⊢ 𝑛 ∈ V
17	16	snnz 4734	. . . . . . 7 ⊢ {𝑛} ≠ ∅
18		ssn0 4357	. . . . . . 7 ⊢ (({𝑛} ⊆ (𝑛 ∪ {𝑛}) ∧ {𝑛} ≠ ∅) → (𝑛 ∪ {𝑛}) ≠ ∅)
19	15, 17, 18	mp2an 693	. . . . . 6 ⊢ (𝑛 ∪ {𝑛}) ≠ ∅
20		neeq1 2995	. . . . . 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 2829	. . . 4 ⊢ (𝑝 ∈ 𝐷 ↔ 𝑝 ∈ (ω ∖ {∅}))
25		eldifsn 4743	. . . 4 ⊢ (𝑝 ∈ (ω ∖ {∅}) ↔ (𝑝 ∈ ω ∧ 𝑝 ≠ ∅))
26	24, 25	bitri 275	. . 3 ⊢ (𝑝 ∈ 𝐷 ↔ (𝑝 ∈ ω ∧ 𝑝 ≠ ∅))
27	12, 23, 26	sylanbrc 584	. 2 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛) → 𝑝 ∈ 𝐷)
28	4, 5, 27	syl2anc 585	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 1542   ∈ wcel 2114   ≠ wne 2933   ∖ cdif 3899   ∪ cun 3900   ⊆ wss 3902  ∅c0 4286  {csn 4581  suc csuc 6320   Fn wfn 6488  ωcom 7810   ∧ w-bnj17 34844   FrSe w-bnj15 34850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-tr 5207  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-om 7811  df-bnj17 34845

This theorem is referenced by:  bnj910  35106