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Mathbox for Scott Fenton |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hfsn | Structured version Visualization version GIF version |
Description: The singleton of an HF set is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.) |
Ref | Expression |
---|---|
hfsn | ⊢ (𝐴 ∈ Hf → {𝐴} ∈ Hf ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ranksng 34683 | . . 3 ⊢ (𝐴 ∈ Hf → (rank‘{𝐴}) = suc (rank‘𝐴)) | |
2 | elhf2g 34692 | . . . . 5 ⊢ (𝐴 ∈ Hf → (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω)) | |
3 | 2 | ibi 267 | . . . 4 ⊢ (𝐴 ∈ Hf → (rank‘𝐴) ∈ ω) |
4 | peano2 7818 | . . . 4 ⊢ ((rank‘𝐴) ∈ ω → suc (rank‘𝐴) ∈ ω) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝐴 ∈ Hf → suc (rank‘𝐴) ∈ ω) |
6 | 1, 5 | eqeltrd 2839 | . 2 ⊢ (𝐴 ∈ Hf → (rank‘{𝐴}) ∈ ω) |
7 | snex 5387 | . . 3 ⊢ {𝐴} ∈ V | |
8 | 7 | elhf2 34691 | . 2 ⊢ ({𝐴} ∈ Hf ↔ (rank‘{𝐴}) ∈ ω) |
9 | 6, 8 | sylibr 233 | 1 ⊢ (𝐴 ∈ Hf → {𝐴} ∈ Hf ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 {csn 4585 suc csuc 6316 ‘cfv 6492 ωcom 7793 rankcrnk 9633 Hf chf 34688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-reg 9462 ax-inf2 9511 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7353 df-om 7794 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-r1 9634 df-rank 9635 df-hf 34689 |
This theorem is referenced by: hfadj 34696 |
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