![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > hfuni | Structured version Visualization version GIF version |
Description: The union of an HF set is itself hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.) |
Ref | Expression |
---|---|
hfuni | ⊢ (𝐴 ∈ Hf → ∪ 𝐴 ∈ Hf ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rankuni 9145 | . . 3 ⊢ (rank‘∪ 𝐴) = ∪ (rank‘𝐴) | |
2 | rankon 9077 | . . . . . 6 ⊢ (rank‘𝐴) ∈ On | |
3 | 2 | ontrci 6178 | . . . . 5 ⊢ Tr (rank‘𝐴) |
4 | df-tr 5071 | . . . . 5 ⊢ (Tr (rank‘𝐴) ↔ ∪ (rank‘𝐴) ⊆ (rank‘𝐴)) | |
5 | 3, 4 | mpbi 231 | . . . 4 ⊢ ∪ (rank‘𝐴) ⊆ (rank‘𝐴) |
6 | elhf2g 33248 | . . . . 5 ⊢ (𝐴 ∈ Hf → (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω)) | |
7 | 6 | ibi 268 | . . . 4 ⊢ (𝐴 ∈ Hf → (rank‘𝐴) ∈ ω) |
8 | rankon 9077 | . . . . . . 7 ⊢ (rank‘∪ 𝐴) ∈ On | |
9 | 1, 8 | eqeltrri 2882 | . . . . . 6 ⊢ ∪ (rank‘𝐴) ∈ On |
10 | 9 | onordi 6177 | . . . . 5 ⊢ Ord ∪ (rank‘𝐴) |
11 | ordom 7452 | . . . . 5 ⊢ Ord ω | |
12 | ordtr2 6117 | . . . . 5 ⊢ ((Ord ∪ (rank‘𝐴) ∧ Ord ω) → ((∪ (rank‘𝐴) ⊆ (rank‘𝐴) ∧ (rank‘𝐴) ∈ ω) → ∪ (rank‘𝐴) ∈ ω)) | |
13 | 10, 11, 12 | mp2an 688 | . . . 4 ⊢ ((∪ (rank‘𝐴) ⊆ (rank‘𝐴) ∧ (rank‘𝐴) ∈ ω) → ∪ (rank‘𝐴) ∈ ω) |
14 | 5, 7, 13 | sylancr 587 | . . 3 ⊢ (𝐴 ∈ Hf → ∪ (rank‘𝐴) ∈ ω) |
15 | 1, 14 | syl5eqel 2889 | . 2 ⊢ (𝐴 ∈ Hf → (rank‘∪ 𝐴) ∈ ω) |
16 | uniexg 7332 | . . 3 ⊢ (𝐴 ∈ Hf → ∪ 𝐴 ∈ V) | |
17 | elhf2g 33248 | . . 3 ⊢ (∪ 𝐴 ∈ V → (∪ 𝐴 ∈ Hf ↔ (rank‘∪ 𝐴) ∈ ω)) | |
18 | 16, 17 | syl 17 | . 2 ⊢ (𝐴 ∈ Hf → (∪ 𝐴 ∈ Hf ↔ (rank‘∪ 𝐴) ∈ ω)) |
19 | 15, 18 | mpbird 258 | 1 ⊢ (𝐴 ∈ Hf → ∪ 𝐴 ∈ Hf ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2083 Vcvv 3440 ⊆ wss 3865 ∪ cuni 4751 Tr wtr 5070 Ord word 6072 Oncon0 6073 ‘cfv 6232 ωcom 7443 rankcrnk 9045 Hf chf 33244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-reg 8909 ax-inf2 8957 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-om 7444 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-r1 9046 df-rank 9047 df-hf 33245 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |