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Mirrors > Home > MPE Home > Th. List > Mathboxes > fveqvfvv | Structured version Visualization version Unicode version |
Description: If a function's value at an argument is the universal class (which can never be the case because of fvex 6201), the function's value at this argument is any set (especially the empty set). In short "If a function's value is a proper class, it is a set", which sounds strange/contradictory, but which is a consequence of that a contradiction implies anything (see pm2.21i 116). (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
fveqvfvv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6201 | . . . 4 | |
2 | eleq1a 2696 | . . . 4 | |
3 | 1, 2 | ax-mp 5 | . . 3 |
4 | vprc 4796 | . . . 4 | |
5 | 4 | pm2.21i 116 | . . 3 |
6 | 3, 5 | syl 17 | . 2 |
7 | 6 | eqcoms 2630 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 cvv 3200 cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-sn 4178 df-pr 4180 df-uni 4437 df-iota 5851 df-fv 5896 |
This theorem is referenced by: afvpcfv0 41226 |
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