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Mirrors > Home > MPE Home > Th. List > Mathboxes > scott0f | Structured version Visualization version GIF version |
Description: A version of scott0 9309 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 9309 | . 2 ⊢ (𝐴 = ∅ ↔ {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = ∅) | |
2 | scott0f.1 | . . . . . 6 ⊢ Ⅎ𝑦𝐴 | |
3 | nfcv 2977 | . . . . . 6 ⊢ Ⅎ𝑧𝐴 | |
4 | nfv 1911 | . . . . . 6 ⊢ Ⅎ𝑧(rank‘𝑥) ⊆ (rank‘𝑦) | |
5 | nfv 1911 | . . . . . 6 ⊢ Ⅎ𝑦(rank‘𝑥) ⊆ (rank‘𝑧) | |
6 | fveq2 6665 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → (rank‘𝑦) = (rank‘𝑧)) | |
7 | 6 | sseq2d 3999 | . . . . . 6 ⊢ (𝑦 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑧))) |
8 | 2, 3, 4, 5, 7 | cbvralfw 3438 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)) |
9 | 8 | rabbii 3474 | . . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)} |
10 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑤𝐴 | |
11 | scott0f.2 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
12 | nfv 1911 | . . . . . 6 ⊢ Ⅎ𝑥(rank‘𝑤) ⊆ (rank‘𝑧) | |
13 | 11, 12 | nfralw 3225 | . . . . 5 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧) |
14 | nfv 1911 | . . . . 5 ⊢ Ⅎ𝑤∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧) | |
15 | fveq2 6665 | . . . . . . 7 ⊢ (𝑤 = 𝑥 → (rank‘𝑤) = (rank‘𝑥)) | |
16 | 15 | sseq1d 3998 | . . . . . 6 ⊢ (𝑤 = 𝑥 → ((rank‘𝑤) ⊆ (rank‘𝑧) ↔ (rank‘𝑥) ⊆ (rank‘𝑧))) |
17 | 16 | ralbidv 3197 | . . . . 5 ⊢ (𝑤 = 𝑥 → (∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧) ↔ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧))) |
18 | 10, 11, 13, 14, 17 | cbvrabw 3490 | . . . 4 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = {𝑥 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑧)} |
19 | 9, 18 | eqtr4i 2847 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} |
20 | 19 | eqeq1i 2826 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅ ↔ {𝑤 ∈ 𝐴 ∣ ∀𝑧 ∈ 𝐴 (rank‘𝑤) ⊆ (rank‘𝑧)} = ∅) |
21 | 1, 20 | bitr4i 280 | 1 ⊢ (𝐴 = ∅ ↔ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 Ⅎwnfc 2961 ∀wral 3138 {crab 3142 ⊆ wss 3936 ∅c0 4291 ‘cfv 6350 rankcrnk 9186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-r1 9187 df-rank 9188 |
This theorem is referenced by: scottn0f 35442 |
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