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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 7590 | . . . 4 ⊢ ∅ ∈ ω | |
2 | peano2 7591 | . . . 4 ⊢ (∅ ∈ ω → suc ∅ ∈ ω) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ suc ∅ ∈ ω |
4 | 0elpw 5247 | . . . 4 ⊢ ∅ ∈ 𝒫 (𝑅1‘∅) | |
5 | 0elon 6237 | . . . . 5 ⊢ ∅ ∈ On | |
6 | r1suc 9187 | . . . . 5 ⊢ (∅ ∈ On → (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅)) | |
7 | 5, 6 | ax-mp 5 | . . . 4 ⊢ (𝑅1‘suc ∅) = 𝒫 (𝑅1‘∅) |
8 | 4, 7 | eleqtrri 2909 | . . 3 ⊢ ∅ ∈ (𝑅1‘suc ∅) |
9 | fveq2 6663 | . . . . 5 ⊢ (𝑥 = suc ∅ → (𝑅1‘𝑥) = (𝑅1‘suc ∅)) | |
10 | 9 | eleq2d 2895 | . . . 4 ⊢ (𝑥 = suc ∅ → (∅ ∈ (𝑅1‘𝑥) ↔ ∅ ∈ (𝑅1‘suc ∅))) |
11 | 10 | rspcev 3620 | . . 3 ⊢ ((suc ∅ ∈ ω ∧ ∅ ∈ (𝑅1‘suc ∅)) → ∃𝑥 ∈ ω ∅ ∈ (𝑅1‘𝑥)) |
12 | 3, 8, 11 | mp2an 688 | . 2 ⊢ ∃𝑥 ∈ ω ∅ ∈ (𝑅1‘𝑥) |
13 | elhf 33532 | . 2 ⊢ (∅ ∈ Hf ↔ ∃𝑥 ∈ ω ∅ ∈ (𝑅1‘𝑥)) | |
14 | 12, 13 | mpbir 232 | 1 ⊢ ∅ ∈ Hf |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 ∃wrex 3136 ∅c0 4288 𝒫 cpw 4535 Oncon0 6184 suc csuc 6186 ‘cfv 6348 ωcom 7569 𝑅1cr1 9179 Hf chf 33530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-r1 9181 df-hf 33531 |
This theorem is referenced by: (None) |
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