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Mirrors > Home > MPE Home > Th. List > Mathboxes > onsetreclem3 | Structured version Visualization version GIF version |
Description: Lemma for onsetrec 42451. (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 5733 | . . . 4 ⊢ (𝑎 ∈ On → Ord 𝑎) | |
2 | orduniorsuc 7030 | . . . 4 ⊢ (Ord 𝑎 → (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝑎 ∈ On → (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎)) |
4 | vex 3203 | . . . 4 ⊢ 𝑎 ∈ V | |
5 | 4 | elpr 4198 | . . 3 ⊢ (𝑎 ∈ {∪ 𝑎, suc ∪ 𝑎} ↔ (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎)) |
6 | 3, 5 | sylibr 224 | . 2 ⊢ (𝑎 ∈ On → 𝑎 ∈ {∪ 𝑎, suc ∪ 𝑎}) |
7 | onsetreclem3.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) | |
8 | 7 | onsetreclem1 42448 | . 2 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎} |
9 | 6, 8 | syl6eleqr 2712 | 1 ⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 383 = wceq 1483 ∈ wcel 1990 Vcvv 3200 {cpr 4179 ∪ cuni 4436 ↦ cmpt 4729 Ord word 5722 Oncon0 5723 suc csuc 5725 ‘cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-ord 5726 df-on 5727 df-suc 5729 df-iota 5851 df-fun 5890 df-fv 5896 |
This theorem is referenced by: onsetrec 42451 |
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