![]() |
Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj98 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj150 31274. 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.) |
Ref | Expression |
---|---|
bnj98 | ⊢ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3343 | . . . . . 6 ⊢ 𝑖 ∈ V | |
2 | 1 | sucid 5965 | . . . . 5 ⊢ 𝑖 ∈ suc 𝑖 |
3 | 2 | n0ii 4065 | . . . 4 ⊢ ¬ suc 𝑖 = ∅ |
4 | df-suc 5890 | . . . . . 6 ⊢ suc 𝑖 = (𝑖 ∪ {𝑖}) | |
5 | df-un 3720 | . . . . . 6 ⊢ (𝑖 ∪ {𝑖}) = {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} | |
6 | 4, 5 | eqtri 2782 | . . . . 5 ⊢ suc 𝑖 = {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} |
7 | df1o2 7743 | . . . . . . 7 ⊢ 1𝑜 = {∅} | |
8 | 6, 7 | eleq12i 2832 | . . . . . 6 ⊢ (suc 𝑖 ∈ 1𝑜 ↔ {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} ∈ {∅}) |
9 | elsni 4338 | . . . . . 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 2806 | . . . 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 3062 | 1 ⊢ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 = wceq 1632 ∈ wcel 2139 {cab 2746 ∀wral 3050 ∪ cun 3713 ∅c0 4058 {csn 4321 ∪ ciun 4672 suc csuc 5886 ‘cfv 6049 ωcom 7231 1𝑜c1o 7723 predc-bnj14 31084 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-v 3342 df-dif 3718 df-un 3720 df-nul 4059 df-sn 4322 df-suc 5890 df-1o 7730 |
This theorem is referenced by: bnj150 31274 |
Copyright terms: Public domain | W3C validator |