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| Mirrors > Home > MPE Home > Th. List > Mathboxes > onsetreclem2 | Structured version Visualization version GIF version | ||
| Description: Lemma for onsetrec 49282. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| onsetreclem2.1 | ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) | 
| Ref | Expression | 
|---|---|
| onsetreclem2 | ⊢ (𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | onsetreclem2.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) | |
| 2 | 1 | onsetreclem1 49279 | . 2 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎} | 
| 3 | vex 3483 | . . . 4 ⊢ 𝑎 ∈ V | |
| 4 | 3 | ssonunii 7802 | . . 3 ⊢ (𝑎 ⊆ On → ∪ 𝑎 ∈ On) | 
| 5 | onsuc 7832 | . . 3 ⊢ (∪ 𝑎 ∈ On → suc ∪ 𝑎 ∈ On) | |
| 6 | prssi 4820 | . . 3 ⊢ ((∪ 𝑎 ∈ On ∧ suc ∪ 𝑎 ∈ On) → {∪ 𝑎, suc ∪ 𝑎} ⊆ On) | |
| 7 | 4, 5, 6 | syl2anc2 585 | . 2 ⊢ (𝑎 ⊆ On → {∪ 𝑎, suc ∪ 𝑎} ⊆ On) | 
| 8 | 2, 7 | eqsstrid 4021 | 1 ⊢ (𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ⊆ wss 3950 {cpr 4627 ∪ cuni 4906 ↦ cmpt 5224 Oncon0 6383 suc csuc 6385 ‘cfv 6560 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-ord 6386 df-on 6387 df-suc 6389 df-iota 6513 df-fun 6562 df-fv 6568 | 
| This theorem is referenced by: onsetrec 49282 | 
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