![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rankr1a | Structured version Visualization version GIF version |
Description: A relationship between rank and 𝑅1, clearly equivalent to ssrankr1 9812 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 9841 for the subset version. (Contributed by Raph Levien, 29-May-2004.) |
Ref | Expression |
---|---|
rankid.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
rankr1a | ⊢ (𝐵 ∈ On → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rankid.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | 1 | ssrankr1 9812 | . . 3 ⊢ (𝐵 ∈ On → (𝐵 ⊆ (rank‘𝐴) ↔ ¬ 𝐴 ∈ (𝑅1‘𝐵))) |
3 | rankon 9772 | . . . 4 ⊢ (rank‘𝐴) ∈ On | |
4 | ontri1 6387 | . . . 4 ⊢ ((𝐵 ∈ On ∧ (rank‘𝐴) ∈ On) → (𝐵 ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ 𝐵)) | |
5 | 3, 4 | mpan2 689 | . . 3 ⊢ (𝐵 ∈ On → (𝐵 ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ 𝐵)) |
6 | 2, 5 | bitr3d 280 | . 2 ⊢ (𝐵 ∈ On → (¬ 𝐴 ∈ (𝑅1‘𝐵) ↔ ¬ (rank‘𝐴) ∈ 𝐵)) |
7 | 6 | con4bid 316 | 1 ⊢ (𝐵 ∈ On → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∈ wcel 2106 Vcvv 3473 ⊆ wss 3944 Oncon0 6353 ‘cfv 6532 𝑅1cr1 9739 rankcrnk 9740 |
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 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-reg 9569 ax-inf2 9618 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-ov 7396 df-om 7839 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-r1 9741 df-rank 9742 |
This theorem is referenced by: r1val2 9814 r1pwALT 9823 elhf2 34977 gruex 42828 |
Copyright terms: Public domain | W3C validator |