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Mirrors > Home > ILE Home > Th. List > ord3ex | Unicode version |
Description: The ordinal number 3 is a set, proved without the Axiom of Union. (Contributed by NM, 2-May-2009.) |
Ref | Expression |
---|---|
ord3ex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 3579 | . 2 | |
2 | pp0ex 4163 | . . . . 5 | |
3 | 2 | pwex 4157 | . . . 4 |
4 | pwprss 3780 | . . . 4 | |
5 | 3, 4 | ssexi 4115 | . . 3 |
6 | snsspr2 3717 | . . . 4 | |
7 | unss2 3289 | . . . 4 | |
8 | 6, 7 | ax-mp 5 | . . 3 |
9 | 5, 8 | ssexi 4115 | . 2 |
10 | 1, 9 | eqeltri 2237 | 1 |
Colors of variables: wff set class |
Syntax hints: wcel 2135 cvv 2722 cun 3110 wss 3112 c0 3405 cpw 3554 csn 3571 cpr 3572 ctp 3573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-nul 4103 ax-pow 4148 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2724 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-pw 3556 df-sn 3577 df-pr 3578 df-tp 3579 |
This theorem is referenced by: (None) |
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