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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 33887. 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 3479 | . . . . . 6 ⊢ 𝑖 ∈ V | |
2 | 1 | sucid 6447 | . . . . 5 ⊢ 𝑖 ∈ suc 𝑖 |
3 | 2 | n0ii 4337 | . . . 4 ⊢ ¬ suc 𝑖 = ∅ |
4 | df-suc 6371 | . . . . . 6 ⊢ suc 𝑖 = (𝑖 ∪ {𝑖}) | |
5 | df-un 3954 | . . . . . 6 ⊢ (𝑖 ∪ {𝑖}) = {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} | |
6 | 4, 5 | eqtri 2761 | . . . . 5 ⊢ suc 𝑖 = {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} |
7 | df1o2 8473 | . . . . . . 7 ⊢ 1o = {∅} | |
8 | 6, 7 | eleq12i 2827 | . . . . . 6 ⊢ (suc 𝑖 ∈ 1o ↔ {𝑥 ∣ (𝑥 ∈ 𝑖 ∨ 𝑥 ∈ {𝑖})} ∈ {∅}) |
9 | elsni 4646 | . . . . . 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 2785 | . . . 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 3066 | 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 846 = wceq 1542 ∈ wcel 2107 {cab 2710 ∀wral 3062 ∪ cun 3947 ∅c0 4323 {csn 4629 ∪ ciun 4998 suc csuc 6367 ‘cfv 6544 ωcom 7855 1oc1o 8459 predc-bnj14 33699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-v 3477 df-dif 3952 df-un 3954 df-nul 4324 df-sn 4630 df-suc 6371 df-1o 8466 |
This theorem is referenced by: bnj150 33887 |
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