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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 7829 | . . . 4 ⊢ ∅ ∈ ω | |
2 | peano2 7831 | . . . 4 ⊢ (∅ ∈ ω → suc ∅ ∈ ω) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ suc ∅ ∈ ω |
4 | 0elpw 5315 | . . . 4 ⊢ ∅ ∈ 𝒫 (𝑅1‘∅) | |
5 | 0elon 6375 | . . . . 5 ⊢ ∅ ∈ On | |
6 | r1suc 9714 | . . . . 5 ⊢ (∅ ∈ On → (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅)) | |
7 | 5, 6 | ax-mp 5 | . . . 4 ⊢ (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅) |
8 | 4, 7 | eleqtrri 2833 | . . 3 ⊢ ∅ ∈ (𝑅1‘suc ∅) |
9 | fveq2 6846 | . . . . 5 ⊢ (𝑥 = suc ∅ → (𝑅1‘𝑥) = (𝑅1‘suc ∅)) | |
10 | 9 | eleq2d 2820 | . . . 4 ⊢ (𝑥 = suc ∅ → (∅ ∈ (𝑅1‘𝑥) ↔ ∅ ∈ (𝑅1‘suc ∅))) |
11 | 10 | rspcev 3583 | . . 3 ⊢ ((suc ∅ ∈ ω ∧ ∅ ∈ (𝑅1‘suc ∅)) → ∃𝑥 ∈ ω ∅ ∈ (𝑅1‘𝑥)) |
12 | 3, 8, 11 | mp2an 691 | . 2 ⊢ ∃𝑥 ∈ ω ∅ ∈ (𝑅1‘𝑥) |
13 | elhf 34812 | . 2 ⊢ (∅ ∈ Hf ↔ ∃𝑥 ∈ ω ∅ ∈ (𝑅1‘𝑥)) | |
14 | 12, 13 | mpbir 230 | 1 ⊢ ∅ ∈ Hf |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ∃wrex 3070 ∅c0 4286 𝒫 cpw 4564 Oncon0 6321 suc csuc 6323 ‘cfv 6500 ωcom 7806 𝑅1cr1 9706 Hf chf 34810 |
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 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-r1 9708 df-hf 34811 |
This theorem is referenced by: (None) |
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