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Mirrors > Home > MPE Home > Th. List > scottexs | Structured version Visualization version GIF version |
Description: Theorem scheme version of scottex 9302. The collection of all 𝑥 of minimum rank such that 𝜑(𝑥) is true, is a set. (Contributed by NM, 13-Oct-2003.) |
Ref | Expression |
---|---|
scottexs | ⊢ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2974 | . . . 4 ⊢ Ⅎ𝑧{𝑥 ∣ 𝜑} | |
2 | nfab1 2976 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
3 | nfv 1906 | . . . . 5 ⊢ Ⅎ𝑥(rank‘𝑧) ⊆ (rank‘𝑦) | |
4 | 2, 3 | nfralw 3222 | . . . 4 ⊢ Ⅎ𝑥∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑧) ⊆ (rank‘𝑦) |
5 | nfv 1906 | . . . 4 ⊢ Ⅎ𝑧∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦) | |
6 | fveq2 6663 | . . . . . 6 ⊢ (𝑧 = 𝑥 → (rank‘𝑧) = (rank‘𝑥)) | |
7 | 6 | sseq1d 3995 | . . . . 5 ⊢ (𝑧 = 𝑥 → ((rank‘𝑧) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑦))) |
8 | 7 | ralbidv 3194 | . . . 4 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑧) ⊆ (rank‘𝑦) ↔ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦))) |
9 | 1, 2, 4, 5, 8 | cbvrabw 3487 | . . 3 ⊢ {𝑧 ∈ {𝑥 ∣ 𝜑} ∣ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)} = {𝑥 ∈ {𝑥 ∣ 𝜑} ∣ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)} |
10 | df-rab 3144 | . . 3 ⊢ {𝑥 ∈ {𝑥 ∣ 𝜑} ∣ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∣ (𝑥 ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦))} | |
11 | abid 2800 | . . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
12 | df-ral 3140 | . . . . . 6 ⊢ (∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → (rank‘𝑥) ⊆ (rank‘𝑦))) | |
13 | df-sbc 3770 | . . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
14 | 13 | imbi1i 351 | . . . . . . 7 ⊢ (([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦 ∈ {𝑥 ∣ 𝜑} → (rank‘𝑥) ⊆ (rank‘𝑦))) |
15 | 14 | albii 1811 | . . . . . 6 ⊢ (∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → (rank‘𝑥) ⊆ (rank‘𝑦))) |
16 | 12, 15 | bitr4i 279 | . . . . 5 ⊢ (∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦))) |
17 | 11, 16 | anbi12i 626 | . . . 4 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))) |
18 | 17 | abbii 2883 | . . 3 ⊢ {𝑥 ∣ (𝑥 ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦))} = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} |
19 | 9, 10, 18 | 3eqtri 2845 | . 2 ⊢ {𝑧 ∈ {𝑥 ∣ 𝜑} ∣ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)} = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} |
20 | scottex 9302 | . 2 ⊢ {𝑧 ∈ {𝑥 ∣ 𝜑} ∣ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)} ∈ V | |
21 | 19, 20 | eqeltrri 2907 | 1 ⊢ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1526 ∈ wcel 2105 {cab 2796 ∀wral 3135 {crab 3139 Vcvv 3492 [wsbc 3769 ⊆ wss 3933 ‘cfv 6348 rankcrnk 9180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-reg 9044 ax-inf2 9092 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-r1 9181 df-rank 9182 |
This theorem is referenced by: hta 9314 |
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