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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 9506 | . . 3 ⊢ (rank‘∪ 𝐴) = ∪ (rank‘𝐴) | |
2 | rankon 9438 | . . . . . 6 ⊢ (rank‘𝐴) ∈ On | |
3 | 2 | ontrci 6339 | . . . . 5 ⊢ Tr (rank‘𝐴) |
4 | df-tr 5178 | . . . . 5 ⊢ (Tr (rank‘𝐴) ↔ ∪ (rank‘𝐴) ⊆ (rank‘𝐴)) | |
5 | 3, 4 | mpbi 233 | . . . 4 ⊢ ∪ (rank‘𝐴) ⊆ (rank‘𝐴) |
6 | elhf2g 34246 | . . . . 5 ⊢ (𝐴 ∈ Hf → (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω)) | |
7 | 6 | ibi 270 | . . . 4 ⊢ (𝐴 ∈ Hf → (rank‘𝐴) ∈ ω) |
8 | rankon 9438 | . . . . . . 7 ⊢ (rank‘∪ 𝐴) ∈ On | |
9 | 1, 8 | eqeltrri 2837 | . . . . . 6 ⊢ ∪ (rank‘𝐴) ∈ On |
10 | 9 | onordi 6338 | . . . . 5 ⊢ Ord ∪ (rank‘𝐴) |
11 | ordom 7675 | . . . . 5 ⊢ Ord ω | |
12 | ordtr2 6277 | . . . . 5 ⊢ ((Ord ∪ (rank‘𝐴) ∧ Ord ω) → ((∪ (rank‘𝐴) ⊆ (rank‘𝐴) ∧ (rank‘𝐴) ∈ ω) → ∪ (rank‘𝐴) ∈ ω)) | |
13 | 10, 11, 12 | mp2an 692 | . . . 4 ⊢ ((∪ (rank‘𝐴) ⊆ (rank‘𝐴) ∧ (rank‘𝐴) ∈ ω) → ∪ (rank‘𝐴) ∈ ω) |
14 | 5, 7, 13 | sylancr 590 | . . 3 ⊢ (𝐴 ∈ Hf → ∪ (rank‘𝐴) ∈ ω) |
15 | 1, 14 | eqeltrid 2844 | . 2 ⊢ (𝐴 ∈ Hf → (rank‘∪ 𝐴) ∈ ω) |
16 | uniexg 7549 | . . 3 ⊢ (𝐴 ∈ Hf → ∪ 𝐴 ∈ V) | |
17 | elhf2g 34246 | . . 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 260 | 1 ⊢ (𝐴 ∈ Hf → ∪ 𝐴 ∈ Hf ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2112 Vcvv 3422 ⊆ wss 3882 ∪ cuni 4835 Tr wtr 5177 Ord word 6232 Oncon0 6233 ‘cfv 6400 ωcom 7665 rankcrnk 9406 Hf chf 34242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-rep 5195 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 ax-reg 9235 ax-inf2 9283 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3711 df-csb 3828 df-dif 3885 df-un 3887 df-in 3889 df-ss 3899 df-pss 3901 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4836 df-int 4876 df-iun 4922 df-br 5070 df-opab 5132 df-mpt 5152 df-tr 5178 df-id 5471 df-eprel 5477 df-po 5485 df-so 5486 df-fr 5526 df-we 5528 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 df-pred 6178 df-ord 6236 df-on 6237 df-lim 6238 df-suc 6239 df-iota 6358 df-fun 6402 df-fn 6403 df-f 6404 df-f1 6405 df-fo 6406 df-f1o 6407 df-fv 6408 df-om 7666 df-wrecs 8070 df-recs 8131 df-rdg 8169 df-er 8414 df-en 8650 df-dom 8651 df-sdom 8652 df-r1 9407 df-rank 9408 df-hf 34243 |
This theorem is referenced by: (None) |
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