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Mirrors > Home > MPE Home > Th. List > Mathboxes > scott0f | Structured version Visualization version GIF version |
Description: A version of scott0 9502 with nonfree 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 9502 | . 2 ⊢ (𝐴 = ∅ ↔ {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = ∅) | |
2 | scott0f.1 | . . . . . 6 ⊢ Ⅎ𝑦𝐴 | |
3 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑧𝐴 | |
4 | nfv 1922 | . . . . . 6 ⊢ Ⅎ𝑧(rank‘𝑥) ⊆ (rank‘𝑦) | |
5 | nfv 1922 | . . . . . 6 ⊢ Ⅎ𝑦(rank‘𝑥) ⊆ (rank‘𝑧) | |
6 | fveq2 6717 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → (rank‘𝑦) = (rank‘𝑧)) | |
7 | 6 | sseq2d 3933 | . . . . . 6 ⊢ (𝑦 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑧))) |
8 | 2, 3, 4, 5, 7 | cbvralfw 3344 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)) |
9 | 8 | rabbii 3383 | . . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)} |
10 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑤𝐴 | |
11 | scott0f.2 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
12 | nfv 1922 | . . . . . 6 ⊢ Ⅎ𝑥(rank‘𝑤) ⊆ (rank‘𝑧) | |
13 | 11, 12 | nfralw 3147 | . . . . 5 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧) |
14 | nfv 1922 | . . . . 5 ⊢ Ⅎ𝑤∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧) | |
15 | fveq2 6717 | . . . . . . 7 ⊢ (𝑤 = 𝑥 → (rank‘𝑤) = (rank‘𝑥)) | |
16 | 15 | sseq1d 3932 | . . . . . 6 ⊢ (𝑤 = 𝑥 → ((rank‘𝑤) ⊆ (rank‘𝑧) ↔ (rank‘𝑥) ⊆ (rank‘𝑧))) |
17 | 16 | ralbidv 3118 | . . . . 5 ⊢ (𝑤 = 𝑥 → (∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧) ↔ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧))) |
18 | 10, 11, 13, 14, 17 | cbvrabw 3400 | . . . 4 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = {𝑥 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)} |
19 | 9, 18 | eqtr4i 2768 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} |
20 | 19 | eqeq1i 2742 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅ ↔ {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = ∅) |
21 | 1, 20 | bitr4i 281 | 1 ⊢ (𝐴 = ∅ ↔ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 Ⅎwnfc 2884 ∀wral 3061 {crab 3065 ⊆ wss 3866 ∅c0 4237 ‘cfv 6380 rankcrnk 9379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-om 7645 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-r1 9380 df-rank 9381 |
This theorem is referenced by: scottn0f 36065 |
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