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| 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 6085 that collapses a proper class into a
set
of minimum rank. The wff |
| Ref | Expression |
|---|---|
| cp |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | visset 2541 |
. . 3
| |
| 2 | 1 | cplem2 6090 |
. 2
|
| 3 | abn0 3099 |
. . . . 5
| |
| 4 | elin 2999 |
. . . . . . . 8
| |
| 5 | abid 2130 |
. . . . . . . . 9
| |
| 6 | 5 | anbi1i 709 |
. . . . . . . 8
|
| 7 | ancom 414 |
. . . . . . . 8
| |
| 8 | 4, 6, 7 | 3bitri 289 |
. . . . . . 7
|
| 9 | 8 | exbii 1687 |
. . . . . 6
|
| 10 | hbab1 2131 |
. . . . . . . 8
| |
| 11 | ax-17 1605 |
. . . . . . . 8
| |
| 12 | 10, 11 | hbin 3012 |
. . . . . . 7
|
| 13 | 12 | n0f 3090 |
. . . . . 6
|
| 14 | df-rex 2360 |
. . . . . 6
| |
| 15 | 9, 13, 14 | 3bitr4i 295 |
. . . . 5
|
| 16 | 3, 15 | imbi12i 301 |
. . . 4
|
| 17 | 16 | ralbii 2377 |
. . 3
|
| 18 | 17 | exbii 1687 |
. 2
|
| 19 | 2, 18 | mpbi 272 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem is referenced by: bnd 6092 |
| This theorem was proved from axioms: ax-1 4 ax-2 5 ax-3 6 ax-mp 7 ax-7 1592 ax-gen 1593 ax-8 1594 ax-9 1595 ax-10 1596 ax-11 1597 ax-12 1598 ax-13 1599 ax-14 1600 ax-17 1605 ax-4 1608 ax-5o 1610 ax-6o 1613 ax-9o 1763 ax-10o 1781 ax-16 1854 ax-11o 1864 ax-ext 2123 ax-rep 3596 ax-sep 3606 ax-nul 3613 ax-pow 3649 ax-pr 3687 ax-un 3929 ax-reg 5928 ax-inf2 5964 |
| This theorem depends on definitions: df-bi 220 df-or 338 df-an 339 df-3or 1103 df-3an 1104 df-ex 1616 df-sb 1816 df-eu 2041 df-mo 2042 df-clab 2129 df-cleq 2134 df-clel 2137 df-ne 2268 df-ral 2359 df-rex 2360 df-rab 2362 df-v 2540 df-sbc 2700 df-csb 2774 df-dif 2830 df-un 2832 df-in 2834 df-ss 2836 df-pss 2838 df-nul 3083 df-if 3181 df-pw 3229 df-sn 3242 df-pr 3243 df-tp 3245 df-op 3246 df-uni 3367 df-int 3401 df-iun 3438 df-iin 3439 df-br 3508 df-opab 3566 df-tr 3580 df-eprel 3744 df-id 3747 df-po 3752 df-so 3764 df-fr 3782 df-we 3798 df-ord 3814 df-on 3815 df-lim 3816 df-suc 3817 df-om 4086 df-xp 4133 df-rel 4134 df-cnv 4135 df-co 4136 df-dm 4137 df-rn 4138 df-res 4139 df-ima 4140 df-fun 4141 df-fn 4142 df-fv 4147 df-rdg 5304 df-r1 5986 df-rank 5987 |