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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 34869. 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 3482 | . . . . . 6 ⊢ 𝑖 ∈ V | |
2 | 1 | sucid 6468 | . . . . 5 ⊢ 𝑖 ∈ suc 𝑖 |
3 | 2 | n0ii 4349 | . . . 4 ⊢ ¬ suc 𝑖 = ∅ |
4 | df-suc 6392 | . . . . . 6 ⊢ suc 𝑖 = (𝑖 ∪ {𝑖}) | |
5 | df-un 3968 | . . . . . 6 ⊢ (𝑖 ∪ {𝑖}) = {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} | |
6 | 4, 5 | eqtri 2763 | . . . . 5 ⊢ suc 𝑖 = {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} |
7 | df1o2 8512 | . . . . . . 7 ⊢ 1o = {∅} | |
8 | 6, 7 | eleq12i 2832 | . . . . . 6 ⊢ (suc 𝑖 ∈ 1o ↔ {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} ∈ {∅}) |
9 | elsni 4648 | . . . . . 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 2787 | . . . 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 3063 | 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 847 = wceq 1537 ∈ wcel 2106 {cab 2712 ∀wral 3059 ∪ cun 3961 ∅c0 4339 {csn 4631 ∪ ciun 4996 suc csuc 6388 ‘cfv 6563 ωcom 7887 1oc1o 8498 predc-bnj14 34681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-v 3480 df-dif 3966 df-un 3968 df-nul 4340 df-sn 4632 df-suc 6392 df-1o 8505 |
This theorem is referenced by: bnj150 34869 |
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