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Mirrors > Home > MPE Home > Th. List > scottexs | Structured version Visualization version GIF version |
Description: Theorem scheme version of scottex 9923. 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 2903 | . . . 4 ⊢ Ⅎ𝑧{𝑥 ∣ 𝜑} | |
2 | nfab1 2905 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
3 | nfv 1912 | . . . . 5 ⊢ Ⅎ𝑥(rank‘𝑧) ⊆ (rank‘𝑦) | |
4 | 2, 3 | nfralw 3309 | . . . 4 ⊢ Ⅎ𝑥∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑧) ⊆ (rank‘𝑦) |
5 | nfv 1912 | . . . 4 ⊢ Ⅎ𝑧∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦) | |
6 | fveq2 6907 | . . . . . 6 ⊢ (𝑧 = 𝑥 → (rank‘𝑧) = (rank‘𝑥)) | |
7 | 6 | sseq1d 4027 | . . . . 5 ⊢ (𝑧 = 𝑥 → ((rank‘𝑧) ⊆ (rank‘𝑦) ↔ (rank‘𝑥) ⊆ (rank‘𝑦))) |
8 | 7 | ralbidv 3176 | . . . 4 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑧) ⊆ (rank‘𝑦) ↔ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦))) |
9 | 1, 2, 4, 5, 8 | cbvrabw 3471 | . . 3 ⊢ {𝑧 ∈ {𝑥 ∣ 𝜑} ∣ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)} = {𝑥 ∈ {𝑥 ∣ 𝜑} ∣ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)} |
10 | df-rab 3434 | . . 3 ⊢ {𝑥 ∈ {𝑥 ∣ 𝜑} ∣ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∣ (𝑥 ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦))} | |
11 | abid 2716 | . . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
12 | df-ral 3060 | . . . . . 6 ⊢ (∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → (rank‘𝑥) ⊆ (rank‘𝑦))) | |
13 | df-sbc 3792 | . . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
14 | 13 | imbi1i 349 | . . . . . . 7 ⊢ (([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦 ∈ {𝑥 ∣ 𝜑} → (rank‘𝑥) ⊆ (rank‘𝑦))) |
15 | 14 | albii 1816 | . . . . . 6 ⊢ (∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} → (rank‘𝑥) ⊆ (rank‘𝑦))) |
16 | 12, 15 | bitr4i 278 | . . . . 5 ⊢ (∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦))) |
17 | 11, 16 | anbi12i 628 | . . . 4 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))) |
18 | 17 | abbii 2807 | . . 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 2767 | . 2 ⊢ {𝑧 ∈ {𝑥 ∣ 𝜑} ∣ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)} = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} |
20 | scottex 9923 | . 2 ⊢ {𝑧 ∈ {𝑥 ∣ 𝜑} ∣ ∀𝑦 ∈ {𝑥 ∣ 𝜑} (rank‘𝑧) ⊆ (rank‘𝑦)} ∈ V | |
21 | 19, 20 | eqeltrri 2836 | 1 ⊢ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1535 ∈ wcel 2106 {cab 2712 ∀wral 3059 {crab 3433 Vcvv 3478 [wsbc 3791 ⊆ wss 3963 ‘cfv 6563 rankcrnk 9801 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-reg 9630 ax-inf2 9679 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-r1 9802 df-rank 9803 |
This theorem is referenced by: hta 9935 |
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