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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 9589 | . . 3 ⊢ ¬ ω ∈ ω | |
2 | elhf2g 35644 | . . . 4 ⊢ (ω ∈ Hf → (ω ∈ Hf ↔ (rank‘ω) ∈ ω)) | |
3 | ordom 7859 | . . . . . . 7 ⊢ Ord ω | |
4 | elong 6363 | . . . . . . 7 ⊢ (ω ∈ Hf → (ω ∈ On ↔ Ord ω)) | |
5 | 3, 4 | mpbiri 258 | . . . . . 6 ⊢ (ω ∈ Hf → ω ∈ On) |
6 | r111 9767 | . . . . . . . . 9 ⊢ 𝑅1:On–1-1→V | |
7 | f1dm 6782 | . . . . . . . . 9 ⊢ (𝑅1:On–1-1→V → dom 𝑅1 = On) | |
8 | 6, 7 | ax-mp 5 | . . . . . . . 8 ⊢ dom 𝑅1 = On |
9 | 8 | eleq2i 2817 | . . . . . . 7 ⊢ (ω ∈ dom 𝑅1 ↔ ω ∈ On) |
10 | rankonid 9821 | . . . . . . 7 ⊢ (ω ∈ dom 𝑅1 ↔ (rank‘ω) = ω) | |
11 | 9, 10 | bitr3i 277 | . . . . . 6 ⊢ (ω ∈ On ↔ (rank‘ω) = ω) |
12 | 5, 11 | sylib 217 | . . . . 5 ⊢ (ω ∈ Hf → (rank‘ω) = ω) |
13 | 12 | eleq1d 2810 | . . . 4 ⊢ (ω ∈ Hf → ((rank‘ω) ∈ ω ↔ ω ∈ ω)) |
14 | 2, 13 | bitrd 279 | . . 3 ⊢ (ω ∈ Hf → (ω ∈ Hf ↔ ω ∈ ω)) |
15 | 1, 14 | mtbiri 327 | . 2 ⊢ (ω ∈ Hf → ¬ ω ∈ Hf ) |
16 | pm2.01 188 | . 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 1533 ∈ wcel 2098 Vcvv 3466 dom cdm 5667 Ord word 6354 Oncon0 6355 –1-1→wf1 6531 ‘cfv 6534 ωcom 7849 𝑅1cr1 9754 rankcrnk 9755 Hf chf 35640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-reg 9584 ax-inf2 9633 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-r1 9756 df-rank 9757 df-hf 35641 |
This theorem is referenced by: (None) |
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