Mathbox for Emmett Weisz |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > onsetreclem3 | Structured version Visualization version GIF version |
Description: Lemma for onsetrec 44895. (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 6187 | . . . 4 ⊢ (𝑎 ∈ On → Ord 𝑎) | |
2 | orduniorsuc 7531 | . . . 4 ⊢ (Ord 𝑎 → (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝑎 ∈ On → (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎)) |
4 | vex 3489 | . . . 4 ⊢ 𝑎 ∈ V | |
5 | 4 | elpr 4576 | . . 3 ⊢ (𝑎 ∈ {∪ 𝑎, suc ∪ 𝑎} ↔ (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎)) |
6 | 3, 5 | sylibr 236 | . 2 ⊢ (𝑎 ∈ On → 𝑎 ∈ {∪ 𝑎, suc ∪ 𝑎}) |
7 | onsetreclem3.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) | |
8 | 7 | onsetreclem1 44892 | . 2 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎} |
9 | 6, 8 | eleqtrrdi 2924 | 1 ⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 = wceq 1537 ∈ wcel 2114 Vcvv 3486 {cpr 4555 ∪ cuni 4824 ↦ cmpt 5132 Ord word 6176 Oncon0 6177 suc csuc 6179 ‘cfv 6341 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pr 5316 ax-un 7447 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-ord 6180 df-on 6181 df-suc 6183 df-iota 6300 df-fun 6343 df-fv 6349 |
This theorem is referenced by: onsetrec 44895 |
Copyright terms: Public domain | W3C validator |