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Mirrors > Home > MPE Home > Th. List > Mathboxes > altopth | Structured version Visualization version Unicode version |
Description: The alternate ordered pair theorem. If two alternate ordered pairs are equal, their first elements are equal and their second elements are equal. Note that and are not required to be a set due to a peculiarity of our specific ordered pair definition, as opposed to the regular ordered pairs used here, which (as in opth 4945), requires to be a set. (Contributed by Scott Fenton, 23-Mar-2012.) |
Ref | Expression |
---|---|
altopth.1 | |
altopth.2 |
Ref | Expression |
---|---|
altopth |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | altopth.1 | . 2 | |
2 | altopth.2 | . 2 | |
3 | altopthg 32074 | . 2 | |
4 | 1, 2, 3 | mp2an 708 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wa 384 wceq 1483 wcel 1990 cvv 3200 caltop 32063 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
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-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-sn 4178 df-pr 4180 df-altop 32065 |
This theorem is referenced by: altopthd 32079 altopelaltxp 32083 |
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