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Mirrors > Home > MPE Home > Th. List > zfcndinf | Structured version Visualization version Unicode version |
Description: Axiom of Infinity ax-inf 8535, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing theorem el 4847 in the proof. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by NM, 15-Aug-2003.) |
Ref | Expression |
---|---|
zfcndinf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | el 4847 | . . 3 | |
2 | nfv 1843 | . . . . . 6 | |
3 | nfe1 2027 | . . . . . . . 8 | |
4 | 2, 3 | nfim 1825 | . . . . . . 7 |
5 | 4 | nfal 2153 | . . . . . 6 |
6 | 2, 5 | nfan 1828 | . . . . 5 |
7 | 6 | nfex 2154 | . . . 4 |
8 | axinfnd 9428 | . . . . 5 | |
9 | 8 | 19.37iv 1911 | . . . 4 |
10 | 7, 9 | exlimi 2086 | . . 3 |
11 | 1, 10 | ax-mp 5 | . 2 |
12 | elequ1 1997 | . . . . . 6 | |
13 | elequ1 1997 | . . . . . . . 8 | |
14 | 13 | anbi1d 741 | . . . . . . 7 |
15 | 14 | exbidv 1850 | . . . . . 6 |
16 | 12, 15 | imbi12d 334 | . . . . 5 |
17 | 16 | cbvalv 2273 | . . . 4 |
18 | 17 | anbi2i 730 | . . 3 |
19 | 18 | exbii 1774 | . 2 |
20 | 11, 19 | mpbir 221 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wal 1481 wceq 1483 wex 1704 wcel 1990 |
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 ax-pow 4843 ax-pr 4906 ax-reg 8497 ax-inf 8535 |
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-ral 2917 df-rex 2918 df-v 3202 df-dif 3577 df-un 3579 df-nul 3916 df-sn 4178 df-pr 4180 |
This theorem is referenced by: (None) |
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