![]() |
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 9906 | . . 3 ⊢ (rank‘∪ 𝐴) = ∪ (rank‘𝐴) | |
2 | rankon 9838 | . . . . . 6 ⊢ (rank‘𝐴) ∈ On | |
3 | ontr 6485 | . . . . . 6 ⊢ ((rank‘𝐴) ∈ On → Tr (rank‘𝐴)) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ Tr (rank‘𝐴) |
5 | df-tr 5271 | . . . . 5 ⊢ (Tr (rank‘𝐴) ↔ ∪ (rank‘𝐴) ⊆ (rank‘𝐴)) | |
6 | 4, 5 | mpbi 229 | . . . 4 ⊢ ∪ (rank‘𝐴) ⊆ (rank‘𝐴) |
7 | elhf2g 36000 | . . . . 5 ⊢ (𝐴 ∈ Hf → (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω)) | |
8 | 7 | ibi 266 | . . . 4 ⊢ (𝐴 ∈ Hf → (rank‘𝐴) ∈ ω) |
9 | rankon 9838 | . . . . . . 7 ⊢ (rank‘∪ 𝐴) ∈ On | |
10 | 1, 9 | eqeltrri 2823 | . . . . . 6 ⊢ ∪ (rank‘𝐴) ∈ On |
11 | 10 | onordi 6487 | . . . . 5 ⊢ Ord ∪ (rank‘𝐴) |
12 | ordom 7886 | . . . . 5 ⊢ Ord ω | |
13 | ordtr2 6420 | . . . . 5 ⊢ ((Ord ∪ (rank‘𝐴) ∧ Ord ω) → ((∪ (rank‘𝐴) ⊆ (rank‘𝐴) ∧ (rank‘𝐴) ∈ ω) → ∪ (rank‘𝐴) ∈ ω)) | |
14 | 11, 12, 13 | mp2an 690 | . . . 4 ⊢ ((∪ (rank‘𝐴) ⊆ (rank‘𝐴) ∧ (rank‘𝐴) ∈ ω) → ∪ (rank‘𝐴) ∈ ω) |
15 | 6, 8, 14 | sylancr 585 | . . 3 ⊢ (𝐴 ∈ Hf → ∪ (rank‘𝐴) ∈ ω) |
16 | 1, 15 | eqeltrid 2830 | . 2 ⊢ (𝐴 ∈ Hf → (rank‘∪ 𝐴) ∈ ω) |
17 | uniexg 7751 | . . 3 ⊢ (𝐴 ∈ Hf → ∪ 𝐴 ∈ V) | |
18 | elhf2g 36000 | . . 3 ⊢ (∪ 𝐴 ∈ V → (∪ 𝐴 ∈ Hf ↔ (rank‘∪ 𝐴) ∈ ω)) | |
19 | 17, 18 | syl 17 | . 2 ⊢ (𝐴 ∈ Hf → (∪ 𝐴 ∈ Hf ↔ (rank‘∪ 𝐴) ∈ ω)) |
20 | 16, 19 | mpbird 256 | 1 ⊢ (𝐴 ∈ Hf → ∪ 𝐴 ∈ Hf ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2099 Vcvv 3462 ⊆ wss 3947 ∪ cuni 4913 Tr wtr 5270 Ord word 6375 Oncon0 6376 ‘cfv 6554 ωcom 7876 rankcrnk 9806 Hf chf 35996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-reg 9635 ax-inf2 9684 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-r1 9807 df-rank 9808 df-hf 35997 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |