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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0hf | Structured version Visualization version GIF version |
Description: The empty set is a hereditarily finite set. (Contributed by Scott Fenton, 9-Jul-2015.) |
Ref | Expression |
---|---|
0hf | ⊢ ∅ ∈ Hf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano1 7817 | . . . 4 ⊢ ∅ ∈ ω | |
2 | peano2 7819 | . . . 4 ⊢ (∅ ∈ ω → suc ∅ ∈ ω) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ suc ∅ ∈ ω |
4 | 0elpw 5309 | . . . 4 ⊢ ∅ ∈ 𝒫 (𝑅1‘∅) | |
5 | 0elon 6369 | . . . . 5 ⊢ ∅ ∈ On | |
6 | r1suc 9664 | . . . . 5 ⊢ (∅ ∈ On → (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅)) | |
7 | 5, 6 | ax-mp 5 | . . . 4 ⊢ (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅) |
8 | 4, 7 | eleqtrri 2837 | . . 3 ⊢ ∅ ∈ (𝑅1‘suc ∅) |
9 | fveq2 6839 | . . . . 5 ⊢ (𝑥 = suc ∅ → (𝑅1‘𝑥) = (𝑅1‘suc ∅)) | |
10 | 9 | eleq2d 2823 | . . . 4 ⊢ (𝑥 = suc ∅ → (∅ ∈ (𝑅1‘𝑥) ↔ ∅ ∈ (𝑅1‘suc ∅))) |
11 | 10 | rspcev 3579 | . . 3 ⊢ ((suc ∅ ∈ ω ∧ ∅ ∈ (𝑅1‘suc ∅)) → ∃𝑥 ∈ ω ∅ ∈ (𝑅1‘𝑥)) |
12 | 3, 8, 11 | mp2an 690 | . 2 ⊢ ∃𝑥 ∈ ω ∅ ∈ (𝑅1‘𝑥) |
13 | elhf 34697 | . 2 ⊢ (∅ ∈ Hf ↔ ∃𝑥 ∈ ω ∅ ∈ (𝑅1‘𝑥)) | |
14 | 12, 13 | mpbir 230 | 1 ⊢ ∅ ∈ Hf |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 ∃wrex 3071 ∅c0 4280 𝒫 cpw 4558 Oncon0 6315 suc csuc 6317 ‘cfv 6493 ωcom 7794 𝑅1cr1 9656 Hf chf 34695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 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 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-om 7795 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-r1 9658 df-hf 34696 |
This theorem is referenced by: (None) |
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