Theorem bnj98 30645

Description: Technical lemma for bnj150 30654. 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.)

Assertion

Ref	Expression
bnj98	⊢ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅))

Proof of Theorem bnj98

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3189	. . . . . 6 ⊢ 𝑖 ∈ V
2	1	sucid 5763	. . . . 5 ⊢ 𝑖 ∈ suc 𝑖
3	2	n0ii 3898	. . . 4 ⊢ ¬ suc 𝑖 = ∅
4		df-suc 5688	. . . . . 6 ⊢ suc 𝑖 = (𝑖 ∪ {𝑖})
5		df-un 3560	. . . . . 6 ⊢ (𝑖 ∪ {𝑖}) = {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})}
6	4, 5	eqtri 2643	. . . . 5 ⊢ suc 𝑖 = {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})}
7		df1o2 7517	. . . . . . 7 ⊢ 1𝑜 = {∅}
8	6, 7	eleq12i 2691	. . . . . 6 ⊢ (suc 𝑖 ∈ 1𝑜 ↔ {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} ∈ {∅})
9		elsni 4165	. . . . . 6 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} ∈ {∅} → {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} = ∅)
10	8, 9	sylbi 207	. . . . 5 ⊢ (suc 𝑖 ∈ 1𝑜 → {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} = ∅)
11	6, 10	syl5eq 2667	. . . 4 ⊢ (suc 𝑖 ∈ 1𝑜 → suc 𝑖 = ∅)
12	3, 11	mto 188	. . 3 ⊢ ¬ suc 𝑖 ∈ 1𝑜
13	12	pm2.21i 116	. 2 ⊢ (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅))
14	13	rgenw 2919	1 ⊢ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅))

Syntax hints:   → wi 4   ∨ wo 383   = wceq 1480   ∈ wcel 1987  {cab 2607  ∀wral 2907   ∪ cun 3553  ∅c0 3891  {csn 4148  ∪ ciun 4485  suc csuc 5684  ‘cfv 5847  ωcom 7012  1_𝑜c1o 7498   predc-bnj14 30461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-v 3188  df-dif 3558  df-un 3560  df-nul 3892  df-sn 4149  df-suc 5688  df-1o 7505

This theorem is referenced by:  bnj150  30654