Mathbox for Giovanni Mascellani |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > scott0f | Structured version Visualization version GIF version |
Description: A version of scott0 8749 with non-free variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.) |
Ref | Expression |
---|---|
scott0f.1 | ⊢ Ⅎ𝑦𝐴 |
scott0f.2 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
scott0f | ⊢ (𝐴 = ∅ ↔ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scott0 8749 | . 2 ⊢ (𝐴 = ∅ ↔ {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = ∅) | |
2 | scott0f.1 | . . . . . . 7 ⊢ Ⅎ𝑦𝐴 | |
3 | nfcv 2764 | . . . . . . 7 ⊢ Ⅎ𝑧𝐴 | |
4 | nfv 1843 | . . . . . . 7 ⊢ Ⅎ𝑧(rank‘𝑥) ⊆ (rank‘𝑦) | |
5 | nfv 1843 | . . . . . . 7 ⊢ Ⅎ𝑦(rank‘𝑥) ⊆ (rank‘𝑧) | |
6 | fveq2 6191 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → (rank‘𝑦) = (rank‘𝑧)) | |
7 | 6 | sseq2d 3633 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑧))) |
8 | 2, 3, 4, 5, 7 | cbvralf 3165 | . . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)) |
9 | 8 | a1i 11 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧))) |
10 | 9 | rabbiia 3185 | . . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)} |
11 | nfcv 2764 | . . . . 5 ⊢ Ⅎ𝑤𝐴 | |
12 | scott0f.2 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
13 | nfv 1843 | . . . . . 6 ⊢ Ⅎ𝑥(rank‘𝑤) ⊆ (rank‘𝑧) | |
14 | 12, 13 | nfral 2945 | . . . . 5 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧) |
15 | nfv 1843 | . . . . 5 ⊢ Ⅎ𝑤∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧) | |
16 | fveq2 6191 | . . . . . . 7 ⊢ (𝑤 = 𝑥 → (rank‘𝑤) = (rank‘𝑥)) | |
17 | 16 | sseq1d 3632 | . . . . . 6 ⊢ (𝑤 = 𝑥 → ((rank‘𝑤) ⊆ (rank‘𝑧) ↔ (rank‘𝑥) ⊆ (rank‘𝑧))) |
18 | 17 | ralbidv 2986 | . . . . 5 ⊢ (𝑤 = 𝑥 → (∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧) ↔ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧))) |
19 | 11, 12, 14, 15, 18 | cbvrab 3198 | . . . 4 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = {𝑥 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)} |
20 | 10, 19 | eqtr4i 2647 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} |
21 | 20 | eqeq1i 2627 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅ ↔ {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = ∅) |
22 | 1, 21 | bitr4i 267 | 1 ⊢ (𝐴 = ∅ ↔ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 ∈ wcel 1990 Ⅎwnfc 2751 ∀wral 2912 {crab 2916 ⊆ wss 3574 ∅c0 3915 ‘cfv 5888 rankcrnk 8626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-r1 8627 df-rank 8628 |
This theorem is referenced by: scottn0f 33978 |
Copyright terms: Public domain | W3C validator |