![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ssrankr1 | Structured version Visualization version GIF version |
Description: A relationship between an ordinal number less than or equal to a rank, and the cumulative hierarchy of sets 𝑅1. Proposition 9.15(3) of [TakeutiZaring] p. 79. (Contributed by NM, 8-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
rankid.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
ssrankr1 | ⊢ (𝐵 ∈ On → (𝐵 ⊆ (rank‘𝐴) ↔ ¬ 𝐴 ∈ (𝑅1‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rankid.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | unir1 9790 | . . . 4 ⊢ ∪ (𝑅1 “ On) = V | |
3 | 1, 2 | eleqtrri 2831 | . . 3 ⊢ 𝐴 ∈ ∪ (𝑅1 “ On) |
4 | r1fnon 9744 | . . . . . 6 ⊢ 𝑅1 Fn On | |
5 | fndm 6641 | . . . . . 6 ⊢ (𝑅1 Fn On → dom 𝑅1 = On) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ dom 𝑅1 = On |
7 | 6 | eleq2i 2824 | . . . 4 ⊢ (𝐵 ∈ dom 𝑅1 ↔ 𝐵 ∈ On) |
8 | 7 | biimpri 227 | . . 3 ⊢ (𝐵 ∈ On → 𝐵 ∈ dom 𝑅1) |
9 | rankr1clem 9797 | . . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (¬ 𝐴 ∈ (𝑅1‘𝐵) ↔ 𝐵 ⊆ (rank‘𝐴))) | |
10 | 3, 8, 9 | sylancr 587 | . 2 ⊢ (𝐵 ∈ On → (¬ 𝐴 ∈ (𝑅1‘𝐵) ↔ 𝐵 ⊆ (rank‘𝐴))) |
11 | 10 | bicomd 222 | 1 ⊢ (𝐵 ∈ On → (𝐵 ⊆ (rank‘𝐴) ↔ ¬ 𝐴 ∈ (𝑅1‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 Vcvv 3473 ⊆ wss 3944 ∪ cuni 4901 dom cdm 5669 “ cima 5672 Oncon0 6353 Fn wfn 6527 ‘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: rankr1a 9813 |
Copyright terms: Public domain | W3C validator |