Mathbox for Emmett Weisz |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > onsetreclem2 | Structured version Visualization version GIF version |
Description: Lemma for onsetrec 44817. (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 44814 | . 2 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎} |
3 | vex 3500 | . . . 4 ⊢ 𝑎 ∈ V | |
4 | 3 | ssonunii 7505 | . . 3 ⊢ (𝑎 ⊆ On → ∪ 𝑎 ∈ On) |
5 | suceloni 7531 | . . 3 ⊢ (∪ 𝑎 ∈ On → suc ∪ 𝑎 ∈ On) | |
6 | prssi 4757 | . . 3 ⊢ ((∪ 𝑎 ∈ On ∧ suc ∪ 𝑎 ∈ On) → {∪ 𝑎, suc ∪ 𝑎} ⊆ On) | |
7 | 4, 5, 6 | syl2anc2 587 | . 2 ⊢ (𝑎 ⊆ On → {∪ 𝑎, suc ∪ 𝑎} ⊆ On) |
8 | 2, 7 | eqsstrid 4018 | 1 ⊢ (𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ⊆ wss 3939 {cpr 4572 ∪ cuni 4841 ↦ cmpt 5149 Oncon0 6194 suc csuc 6196 ‘cfv 6358 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-ord 6197 df-on 6198 df-suc 6200 df-iota 6317 df-fun 6360 df-fv 6366 |
This theorem is referenced by: onsetrec 44817 |
Copyright terms: Public domain | W3C validator |