Theorem bnj986 35137

Description: Technical lemma for bnj69 35192. 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
bnj986.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj986.10	⊢ 𝐷 = (ω ∖ {∅})
bnj986.15	⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))

Assertion

Ref	Expression
bnj986	⊢ (𝜒 → ∃𝑚∃𝑝𝜏)

Distinct variable group:   𝑚,𝑛,𝑝

Allowed substitution hints:   𝜑(𝑓,𝑚,𝑛,𝑝)   𝜓(𝑓,𝑚,𝑛,𝑝)   𝜒(𝑓,𝑚,𝑛,𝑝)   𝜏(𝑓,𝑚,𝑛,𝑝)   𝐷(𝑓,𝑚,𝑛,𝑝)

Proof of Theorem bnj986

Step	Hyp	Ref	Expression
1		bnj986.3	. . . . . 6 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
2		bnj986.10	. . . . . . 7 ⊢ 𝐷 = (ω ∖ {∅})
3	2	bnj158 34912	. . . . . 6 ⊢ (𝑛 ∈ 𝐷 → ∃𝑚 ∈ ω 𝑛 = suc 𝑚)
4	1, 3	bnj769 34945	. . . . 5 ⊢ (𝜒 → ∃𝑚 ∈ ω 𝑛 = suc 𝑚)
5	4	bnj1196 34976	. . . 4 ⊢ (𝜒 → ∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚))
6		vex 3446	. . . . . 6 ⊢ 𝑛 ∈ V
7	6	sucex 7763	. . . . 5 ⊢ suc 𝑛 ∈ V
8	7	isseti 3460	. . . 4 ⊢ ∃𝑝 𝑝 = suc 𝑛
9	5, 8	jctir 520	. . 3 ⊢ (𝜒 → (∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛))
10		exdistr 1956	. . . 4 ⊢ (∃𝑚∃𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛) ↔ ∃𝑚((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛))
11		19.41v 1951	. . . 4 ⊢ (∃𝑚((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛) ↔ (∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛))
12	10, 11	bitr2i 276	. . 3 ⊢ ((∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛) ↔ ∃𝑚∃𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
13	9, 12	sylib 218	. 2 ⊢ (𝜒 → ∃𝑚∃𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
14		bnj986.15	. . . 4 ⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
15		df-3an 1089	. . . 4 ⊢ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ↔ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
16	14, 15	bitri 275	. . 3 ⊢ (𝜏 ↔ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
17	16	2exbii 1851	. 2 ⊢ (∃𝑚∃𝑝𝜏 ↔ ∃𝑚∃𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
18	13, 17	sylibr 234	1 ⊢ (𝜒 → ∃𝑚∃𝑝𝜏)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3062   ∖ cdif 3900  ∅c0 4287  {csn 4582  suc csuc 6329   Fn wfn 6497  ωcom 7820   ∧ w-bnj17 34869

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 5245  ax-nul 5255  ax-pr 5381  ax-un 7692

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 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-om 7821  df-bnj17 34870

This theorem is referenced by:  bnj996  35138