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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 9532 | . . 3 ⊢ ¬ ω ∈ ω | |
2 | elhf2g 34752 | . . . 4 ⊢ (ω ∈ Hf → (ω ∈ Hf ↔ (rank‘ω) ∈ ω)) | |
3 | ordom 7811 | . . . . . . 7 ⊢ Ord ω | |
4 | elong 6325 | . . . . . . 7 ⊢ (ω ∈ Hf → (ω ∈ On ↔ Ord ω)) | |
5 | 3, 4 | mpbiri 257 | . . . . . 6 ⊢ (ω ∈ Hf → ω ∈ On) |
6 | r111 9710 | . . . . . . . . 9 ⊢ 𝑅1:On–1-1→V | |
7 | f1dm 6742 | . . . . . . . . 9 ⊢ (𝑅1:On–1-1→V → dom 𝑅1 = On) | |
8 | 6, 7 | ax-mp 5 | . . . . . . . 8 ⊢ dom 𝑅1 = On |
9 | 8 | eleq2i 2829 | . . . . . . 7 ⊢ (ω ∈ dom 𝑅1 ↔ ω ∈ On) |
10 | rankonid 9764 | . . . . . . 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 2822 | . . . 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 3445 dom cdm 5633 Ord word 6316 Oncon0 6317 –1-1→wf1 6493 ‘cfv 6496 ωcom 7801 𝑅1cr1 9697 rankcrnk 9698 Hf chf 34748 |
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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-reg 9527 ax-inf2 9576 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7359 df-om 7802 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-er 8647 df-en 8883 df-dom 8884 df-sdom 8885 df-r1 9699 df-rank 9700 df-hf 34749 |
This theorem is referenced by: (None) |
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