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Mirrors > Home > MPE Home > Th. List > ssoprab2b | Structured version Visualization version Unicode version |
Description: Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 5002. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
Ref | Expression |
---|---|
ssoprab2b |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfoprab1 6704 | . . . 4 | |
2 | nfoprab1 6704 | . . . 4 | |
3 | 1, 2 | nfss 3596 | . . 3 |
4 | nfoprab2 6705 | . . . . 5 | |
5 | nfoprab2 6705 | . . . . 5 | |
6 | 4, 5 | nfss 3596 | . . . 4 |
7 | nfoprab3 6706 | . . . . . 6 | |
8 | nfoprab3 6706 | . . . . . 6 | |
9 | 7, 8 | nfss 3596 | . . . . 5 |
10 | ssel 3597 | . . . . . 6 | |
11 | oprabid 6677 | . . . . . 6 | |
12 | oprabid 6677 | . . . . . 6 | |
13 | 10, 11, 12 | 3imtr3g 284 | . . . . 5 |
14 | 9, 13 | alrimi 2082 | . . . 4 |
15 | 6, 14 | alrimi 2082 | . . 3 |
16 | 3, 15 | alrimi 2082 | . 2 |
17 | ssoprab2 6711 | . 2 | |
18 | 16, 17 | impbii 199 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wal 1481 wcel 1990 wss 3574 cop 4183 coprab 6651 |
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-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-oprab 6654 |
This theorem is referenced by: eqoprab2b 6713 |
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