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Mirrors > Home > MPE Home > Th. List > Mathboxes > onsetreclem3 | Structured version Visualization version GIF version |
Description: Lemma for onsetrec 46084. (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 6223 | . . . 4 ⊢ (𝑎 ∈ On → Ord 𝑎) | |
2 | orduniorsuc 7609 | . . . 4 ⊢ (Ord 𝑎 → (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝑎 ∈ On → (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎)) |
4 | vex 3412 | . . . 4 ⊢ 𝑎 ∈ V | |
5 | 4 | elpr 4564 | . . 3 ⊢ (𝑎 ∈ {∪ 𝑎, suc ∪ 𝑎} ↔ (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎)) |
6 | 3, 5 | sylibr 237 | . 2 ⊢ (𝑎 ∈ On → 𝑎 ∈ {∪ 𝑎, suc ∪ 𝑎}) |
7 | onsetreclem3.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) | |
8 | 7 | onsetreclem1 46081 | . 2 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎} |
9 | 6, 8 | eleqtrrdi 2849 | 1 ⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 847 = wceq 1543 ∈ wcel 2110 Vcvv 3408 {cpr 4543 ∪ cuni 4819 ↦ cmpt 5135 Ord word 6212 Oncon0 6213 suc csuc 6215 ‘cfv 6380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-ord 6216 df-on 6217 df-suc 6219 df-iota 6338 df-fun 6382 df-fv 6388 |
This theorem is referenced by: onsetrec 46084 |
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