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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 9591 | . . 3 ⊢ ¬ ω ∈ ω | |
2 | elhf2g 35143 | . . . 4 ⊢ (ω ∈ Hf → (ω ∈ Hf ↔ (rank‘ω) ∈ ω)) | |
3 | ordom 7864 | . . . . . . 7 ⊢ Ord ω | |
4 | elong 6372 | . . . . . . 7 ⊢ (ω ∈ Hf → (ω ∈ On ↔ Ord ω)) | |
5 | 3, 4 | mpbiri 257 | . . . . . 6 ⊢ (ω ∈ Hf → ω ∈ On) |
6 | r111 9769 | . . . . . . . . 9 ⊢ 𝑅1:On–1-1→V | |
7 | f1dm 6791 | . . . . . . . . 9 ⊢ (𝑅1:On–1-1→V → dom 𝑅1 = On) | |
8 | 6, 7 | ax-mp 5 | . . . . . . . 8 ⊢ dom 𝑅1 = On |
9 | 8 | eleq2i 2825 | . . . . . . 7 ⊢ (ω ∈ dom 𝑅1 ↔ ω ∈ On) |
10 | rankonid 9823 | . . . . . . 7 ⊢ (ω ∈ dom 𝑅1 ↔ (rank‘ω) = ω) | |
11 | 9, 10 | bitr3i 276 | . . . . . 6 ⊢ (ω ∈ On ↔ (rank‘ω) = ω) |
12 | 5, 11 | sylib 217 | . . . . 5 ⊢ (ω ∈ Hf → (rank‘ω) = ω) |
13 | 12 | eleq1d 2818 | . . . 4 ⊢ (ω ∈ Hf → ((rank‘ω) ∈ ω ↔ ω ∈ ω)) |
14 | 2, 13 | bitrd 278 | . . 3 ⊢ (ω ∈ Hf → (ω ∈ Hf ↔ ω ∈ ω)) |
15 | 1, 14 | mtbiri 326 | . 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 1541 ∈ wcel 2106 Vcvv 3474 dom cdm 5676 Ord word 6363 Oncon0 6364 –1-1→wf1 6540 ‘cfv 6543 ωcom 7854 𝑅1cr1 9756 rankcrnk 9757 Hf chf 35139 |
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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-reg 9586 ax-inf2 9635 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-om 7855 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-r1 9758 df-rank 9759 df-hf 35140 |
This theorem is referenced by: (None) |
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