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| Mirrors > Home > MPE Home > Th. List > Mathboxes > onsetreclem3 | Structured version Visualization version GIF version | ||
| Description: Lemma for onsetrec 49227. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| onsetreclem3.1 | ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) | 
| Ref | Expression | 
|---|---|
| onsetreclem3 | ⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eloni 6394 | . . . 4 ⊢ (𝑎 ∈ On → Ord 𝑎) | |
| 2 | orduniorsuc 7850 | . . . 4 ⊢ (Ord 𝑎 → (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎)) | |
| 3 | 1, 2 | syl 17 | . . 3 ⊢ (𝑎 ∈ On → (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎)) | 
| 4 | vex 3484 | . . . 4 ⊢ 𝑎 ∈ V | |
| 5 | 4 | elpr 4650 | . . 3 ⊢ (𝑎 ∈ {∪ 𝑎, suc ∪ 𝑎} ↔ (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎)) | 
| 6 | 3, 5 | sylibr 234 | . 2 ⊢ (𝑎 ∈ On → 𝑎 ∈ {∪ 𝑎, suc ∪ 𝑎}) | 
| 7 | onsetreclem3.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) | |
| 8 | 7 | onsetreclem1 49224 | . 2 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎} | 
| 9 | 6, 8 | eleqtrrdi 2852 | 1 ⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∨ wo 848 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {cpr 4628 ∪ cuni 4907 ↦ cmpt 5225 Ord word 6383 Oncon0 6384 suc csuc 6386 ‘cfv 6561 | 
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fv 6569 | 
| This theorem is referenced by: onsetrec 49227 | 
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