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Mirrors > Home > MPE Home > Th. List > cp | Structured version Visualization version GIF version |
Description: Collection Principle. This remarkable theorem scheme is in effect a very strong generalization of the Axiom of Replacement. The proof makes use of Scott's trick scottex 9466 that collapses a proper class into a set of minimum rank. The wff 𝜑 can be thought of as 𝜑(𝑥, 𝑦). Scheme "Collection Principle" of [Jech] p. 72. (Contributed by NM, 17-Oct-2003.) |
Ref | Expression |
---|---|
cp | ⊢ ∃𝑤∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → ∃𝑦 ∈ 𝑤 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3402 | . . 3 ⊢ 𝑧 ∈ V | |
2 | 1 | cplem2 9471 | . 2 ⊢ ∃𝑤∀𝑥 ∈ 𝑧 ({𝑦 ∣ 𝜑} ≠ ∅ → ({𝑦 ∣ 𝜑} ∩ 𝑤) ≠ ∅) |
3 | abn0 4281 | . . . . 5 ⊢ ({𝑦 ∣ 𝜑} ≠ ∅ ↔ ∃𝑦𝜑) | |
4 | elin 3869 | . . . . . . . 8 ⊢ (𝑦 ∈ ({𝑦 ∣ 𝜑} ∩ 𝑤) ↔ (𝑦 ∈ {𝑦 ∣ 𝜑} ∧ 𝑦 ∈ 𝑤)) | |
5 | abid 2718 | . . . . . . . . 9 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑) | |
6 | 5 | anbi1i 627 | . . . . . . . 8 ⊢ ((𝑦 ∈ {𝑦 ∣ 𝜑} ∧ 𝑦 ∈ 𝑤) ↔ (𝜑 ∧ 𝑦 ∈ 𝑤)) |
7 | ancom 464 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑤) ↔ (𝑦 ∈ 𝑤 ∧ 𝜑)) | |
8 | 4, 6, 7 | 3bitri 300 | . . . . . . 7 ⊢ (𝑦 ∈ ({𝑦 ∣ 𝜑} ∩ 𝑤) ↔ (𝑦 ∈ 𝑤 ∧ 𝜑)) |
9 | 8 | exbii 1855 | . . . . . 6 ⊢ (∃𝑦 𝑦 ∈ ({𝑦 ∣ 𝜑} ∩ 𝑤) ↔ ∃𝑦(𝑦 ∈ 𝑤 ∧ 𝜑)) |
10 | nfab1 2899 | . . . . . . . 8 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑} | |
11 | nfcv 2897 | . . . . . . . 8 ⊢ Ⅎ𝑦𝑤 | |
12 | 10, 11 | nfin 4117 | . . . . . . 7 ⊢ Ⅎ𝑦({𝑦 ∣ 𝜑} ∩ 𝑤) |
13 | 12 | n0f 4243 | . . . . . 6 ⊢ (({𝑦 ∣ 𝜑} ∩ 𝑤) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ({𝑦 ∣ 𝜑} ∩ 𝑤)) |
14 | df-rex 3057 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝑤 𝜑 ↔ ∃𝑦(𝑦 ∈ 𝑤 ∧ 𝜑)) | |
15 | 9, 13, 14 | 3bitr4i 306 | . . . . 5 ⊢ (({𝑦 ∣ 𝜑} ∩ 𝑤) ≠ ∅ ↔ ∃𝑦 ∈ 𝑤 𝜑) |
16 | 3, 15 | imbi12i 354 | . . . 4 ⊢ (({𝑦 ∣ 𝜑} ≠ ∅ → ({𝑦 ∣ 𝜑} ∩ 𝑤) ≠ ∅) ↔ (∃𝑦𝜑 → ∃𝑦 ∈ 𝑤 𝜑)) |
17 | 16 | ralbii 3078 | . . 3 ⊢ (∀𝑥 ∈ 𝑧 ({𝑦 ∣ 𝜑} ≠ ∅ → ({𝑦 ∣ 𝜑} ∩ 𝑤) ≠ ∅) ↔ ∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → ∃𝑦 ∈ 𝑤 𝜑)) |
18 | 17 | exbii 1855 | . 2 ⊢ (∃𝑤∀𝑥 ∈ 𝑧 ({𝑦 ∣ 𝜑} ≠ ∅ → ({𝑦 ∣ 𝜑} ∩ 𝑤) ≠ ∅) ↔ ∃𝑤∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → ∃𝑦 ∈ 𝑤 𝜑)) |
19 | 2, 18 | mpbi 233 | 1 ⊢ ∃𝑤∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → ∃𝑦 ∈ 𝑤 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∃wex 1787 ∈ wcel 2112 {cab 2714 ≠ wne 2932 ∀wral 3051 ∃wrex 3052 ∩ cin 3852 ∅c0 4223 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-reg 9186 ax-inf2 9234 |
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 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-r1 9345 df-rank 9346 |
This theorem is referenced by: bnd 9473 |
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