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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj98 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj150 33155. 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 3445 | . . . . . 6 ⊢ 𝑖 ∈ V | |
2 | 1 | sucid 6383 | . . . . 5 ⊢ 𝑖 ∈ suc 𝑖 |
3 | 2 | n0ii 4283 | . . . 4 ⊢ ¬ suc 𝑖 = ∅ |
4 | df-suc 6308 | . . . . . 6 ⊢ suc 𝑖 = (𝑖 ∪ {𝑖}) | |
5 | df-un 3903 | . . . . . 6 ⊢ (𝑖 ∪ {𝑖}) = {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} | |
6 | 4, 5 | eqtri 2764 | . . . . 5 ⊢ suc 𝑖 = {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} |
7 | df1o2 8374 | . . . . . . 7 ⊢ 1o = {∅} | |
8 | 6, 7 | eleq12i 2829 | . . . . . 6 ⊢ (suc 𝑖 ∈ 1o ↔ {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} ∈ {∅}) |
9 | elsni 4590 | . . . . . 6 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} ∈ {∅} → {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} = ∅) | |
10 | 8, 9 | sylbi 216 | . . . . 5 ⊢ (suc 𝑖 ∈ 1o → {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} = ∅) |
11 | 6, 10 | eqtrid 2788 | . . . 4 ⊢ (suc 𝑖 ∈ 1o → suc 𝑖 = ∅) |
12 | 3, 11 | mto 196 | . . 3 ⊢ ¬ suc 𝑖 ∈ 1o |
13 | 12 | pm2.21i 119 | . 2 ⊢ (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)) |
14 | 13 | rgenw 3065 | 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 844 = wceq 1540 ∈ wcel 2105 {cab 2713 ∀wral 3061 ∪ cun 3896 ∅c0 4269 {csn 4573 ∪ ciun 4941 suc csuc 6304 ‘cfv 6479 ωcom 7780 1oc1o 8360 predc-bnj14 32967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3062 df-v 3443 df-dif 3901 df-un 3903 df-nul 4270 df-sn 4574 df-suc 6308 df-1o 8367 |
This theorem is referenced by: bnj150 33155 |
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