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Mirrors > Home > MPE Home > Th. List > Mathboxes > onsetreclem3 | Structured version Visualization version GIF version |
Description: Lemma for onsetrec 45237. (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 6169 | . . . 4 ⊢ (𝑎 ∈ On → Ord 𝑎) | |
2 | orduniorsuc 7525 | . . . 4 ⊢ (Ord 𝑎 → (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝑎 ∈ On → (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎)) |
4 | vex 3444 | . . . 4 ⊢ 𝑎 ∈ V | |
5 | 4 | elpr 4548 | . . 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 45234 | . 2 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎} |
9 | 6, 8 | eleqtrrdi 2901 | 1 ⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 844 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {cpr 4527 ∪ cuni 4800 ↦ cmpt 5110 Ord word 6158 Oncon0 6159 suc csuc 6161 ‘cfv 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-ord 6162 df-on 6163 df-suc 6165 df-iota 6283 df-fun 6326 df-fv 6332 |
This theorem is referenced by: onsetrec 45237 |
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