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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 9286 | . . 3 ⊢ (rank‘∪ 𝐴) = ∪ (rank‘𝐴) | |
2 | rankon 9218 | . . . . . 6 ⊢ (rank‘𝐴) ∈ On | |
3 | 2 | ontrci 6290 | . . . . 5 ⊢ Tr (rank‘𝐴) |
4 | df-tr 5165 | . . . . 5 ⊢ (Tr (rank‘𝐴) ↔ ∪ (rank‘𝐴) ⊆ (rank‘𝐴)) | |
5 | 3, 4 | mpbi 232 | . . . 4 ⊢ ∪ (rank‘𝐴) ⊆ (rank‘𝐴) |
6 | elhf2g 33632 | . . . . 5 ⊢ (𝐴 ∈ Hf → (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω)) | |
7 | 6 | ibi 269 | . . . 4 ⊢ (𝐴 ∈ Hf → (rank‘𝐴) ∈ ω) |
8 | rankon 9218 | . . . . . . 7 ⊢ (rank‘∪ 𝐴) ∈ On | |
9 | 1, 8 | eqeltrri 2910 | . . . . . 6 ⊢ ∪ (rank‘𝐴) ∈ On |
10 | 9 | onordi 6289 | . . . . 5 ⊢ Ord ∪ (rank‘𝐴) |
11 | ordom 7583 | . . . . 5 ⊢ Ord ω | |
12 | ordtr2 6229 | . . . . 5 ⊢ ((Ord ∪ (rank‘𝐴) ∧ Ord ω) → ((∪ (rank‘𝐴) ⊆ (rank‘𝐴) ∧ (rank‘𝐴) ∈ ω) → ∪ (rank‘𝐴) ∈ ω)) | |
13 | 10, 11, 12 | mp2an 690 | . . . 4 ⊢ ((∪ (rank‘𝐴) ⊆ (rank‘𝐴) ∧ (rank‘𝐴) ∈ ω) → ∪ (rank‘𝐴) ∈ ω) |
14 | 5, 7, 13 | sylancr 589 | . . 3 ⊢ (𝐴 ∈ Hf → ∪ (rank‘𝐴) ∈ ω) |
15 | 1, 14 | eqeltrid 2917 | . 2 ⊢ (𝐴 ∈ Hf → (rank‘∪ 𝐴) ∈ ω) |
16 | uniexg 7460 | . . 3 ⊢ (𝐴 ∈ Hf → ∪ 𝐴 ∈ V) | |
17 | elhf2g 33632 | . . 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 259 | 1 ⊢ (𝐴 ∈ Hf → ∪ 𝐴 ∈ Hf ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 ∪ cuni 4831 Tr wtr 5164 Ord word 6184 Oncon0 6185 ‘cfv 6349 ωcom 7574 rankcrnk 9186 Hf chf 33628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-reg 9050 ax-inf2 9098 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-r1 9187 df-rank 9188 df-hf 33629 |
This theorem is referenced by: (None) |
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