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Mirrors > Home > MPE Home > Th. List > Mathboxes > hfninf | Structured version Visualization version GIF version |
Description: ω is not hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.) |
Ref | Expression |
---|---|
hfninf | ⊢ ¬ ω ∈ Hf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elirr 8854 | . . 3 ⊢ ¬ ω ∈ ω | |
2 | elhf2g 33195 | . . . 4 ⊢ (ω ∈ Hf → (ω ∈ Hf ↔ (rank‘ω) ∈ ω)) | |
3 | ordom 7403 | . . . . . . 7 ⊢ Ord ω | |
4 | elong 6034 | . . . . . . 7 ⊢ (ω ∈ Hf → (ω ∈ On ↔ Ord ω)) | |
5 | 3, 4 | mpbiri 250 | . . . . . 6 ⊢ (ω ∈ Hf → ω ∈ On) |
6 | r111 8996 | . . . . . . . . 9 ⊢ 𝑅1:On–1-1→V | |
7 | f1dm 6405 | . . . . . . . . 9 ⊢ (𝑅1:On–1-1→V → dom 𝑅1 = On) | |
8 | 6, 7 | ax-mp 5 | . . . . . . . 8 ⊢ dom 𝑅1 = On |
9 | 8 | eleq2i 2850 | . . . . . . 7 ⊢ (ω ∈ dom 𝑅1 ↔ ω ∈ On) |
10 | rankonid 9050 | . . . . . . 7 ⊢ (ω ∈ dom 𝑅1 ↔ (rank‘ω) = ω) | |
11 | 9, 10 | bitr3i 269 | . . . . . 6 ⊢ (ω ∈ On ↔ (rank‘ω) = ω) |
12 | 5, 11 | sylib 210 | . . . . 5 ⊢ (ω ∈ Hf → (rank‘ω) = ω) |
13 | 12 | eleq1d 2843 | . . . 4 ⊢ (ω ∈ Hf → ((rank‘ω) ∈ ω ↔ ω ∈ ω)) |
14 | 2, 13 | bitrd 271 | . . 3 ⊢ (ω ∈ Hf → (ω ∈ Hf ↔ ω ∈ ω)) |
15 | 1, 14 | mtbiri 319 | . 2 ⊢ (ω ∈ Hf → ¬ ω ∈ Hf ) |
16 | pm2.01 181 | . 2 ⊢ ((ω ∈ Hf → ¬ ω ∈ Hf ) → ¬ ω ∈ Hf ) | |
17 | 15, 16 | ax-mp 5 | 1 ⊢ ¬ ω ∈ Hf |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1508 ∈ wcel 2051 Vcvv 3408 dom cdm 5403 Ord word 6025 Oncon0 6026 –1-1→wf1 6182 ‘cfv 6185 ωcom 7394 𝑅1cr1 8983 rankcrnk 8984 Hf chf 33191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-reg 8849 ax-inf2 8896 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-om 7395 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-r1 8985 df-rank 8986 df-hf 33192 |
This theorem is referenced by: (None) |
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