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 32150. 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 3499 | . . . . . 6 ⊢ 𝑖 ∈ V | |
2 | 1 | sucid 6272 | . . . . 5 ⊢ 𝑖 ∈ suc 𝑖 |
3 | 2 | n0ii 4304 | . . . 4 ⊢ ¬ suc 𝑖 = ∅ |
4 | df-suc 6199 | . . . . . 6 ⊢ suc 𝑖 = (𝑖 ∪ {𝑖}) | |
5 | df-un 3943 | . . . . . 6 ⊢ (𝑖 ∪ {𝑖}) = {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} | |
6 | 4, 5 | eqtri 2846 | . . . . 5 ⊢ suc 𝑖 = {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} |
7 | df1o2 8118 | . . . . . . 7 ⊢ 1o = {∅} | |
8 | 6, 7 | eleq12i 2907 | . . . . . 6 ⊢ (suc 𝑖 ∈ 1o ↔ {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} ∈ {∅}) |
9 | elsni 4586 | . . . . . 6 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} ∈ {∅} → {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} = ∅) | |
10 | 8, 9 | sylbi 219 | . . . . 5 ⊢ (suc 𝑖 ∈ 1o → {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} = ∅) |
11 | 6, 10 | syl5eq 2870 | . . . 4 ⊢ (suc 𝑖 ∈ 1o → suc 𝑖 = ∅) |
12 | 3, 11 | mto 199 | . . 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 3152 | 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 843 = wceq 1537 ∈ wcel 2114 {cab 2801 ∀wral 3140 ∪ cun 3936 ∅c0 4293 {csn 4569 ∪ ciun 4921 suc csuc 6195 ‘cfv 6357 ωcom 7582 1oc1o 8097 predc-bnj14 31960 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-v 3498 df-dif 3941 df-un 3943 df-nul 4294 df-sn 4570 df-suc 6199 df-1o 8104 |
This theorem is referenced by: bnj150 32150 |
Copyright terms: Public domain | W3C validator |