| Mathbox for Jonathan Ben-Naim |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj98 | Structured version Visualization version GIF version | ||
| Description: Technical lemma for bnj150 34890. 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 3484 | . . . . . 6 ⊢ 𝑖 ∈ V | |
| 2 | 1 | sucid 6466 | . . . . 5 ⊢ 𝑖 ∈ suc 𝑖 |
| 3 | 2 | n0ii 4343 | . . . 4 ⊢ ¬ suc 𝑖 = ∅ |
| 4 | df-suc 6390 | . . . . . 6 ⊢ suc 𝑖 = (𝑖 ∪ {𝑖}) | |
| 5 | df-un 3956 | . . . . . 6 ⊢ (𝑖 ∪ {𝑖}) = {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} | |
| 6 | 4, 5 | eqtri 2765 | . . . . 5 ⊢ suc 𝑖 = {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} |
| 7 | df1o2 8513 | . . . . . . 7 ⊢ 1o = {∅} | |
| 8 | 6, 7 | eleq12i 2834 | . . . . . 6 ⊢ (suc 𝑖 ∈ 1o ↔ {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} ∈ {∅}) |
| 9 | elsni 4643 | . . . . . 6 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} ∈ {∅} → {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} = ∅) | |
| 10 | 8, 9 | sylbi 217 | . . . . 5 ⊢ (suc 𝑖 ∈ 1o → {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} = ∅) |
| 11 | 6, 10 | eqtrid 2789 | . . . 4 ⊢ (suc 𝑖 ∈ 1o → suc 𝑖 = ∅) |
| 12 | 3, 11 | mto 197 | . . 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 848 = wceq 1540 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∪ cun 3949 ∅c0 4333 {csn 4626 ∪ ciun 4991 suc csuc 6386 ‘cfv 6561 ωcom 7887 1oc1o 8499 predc-bnj14 34702 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-v 3482 df-dif 3954 df-un 3956 df-nul 4334 df-sn 4627 df-suc 6390 df-1o 8506 |
| This theorem is referenced by: bnj150 34890 |
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