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Mirrors > Home > MPE Home > Th. List > 0we1 | Structured version Visualization version GIF version |
Description: The empty set is a well-ordering of ordinal one. (Contributed by Mario Carneiro, 9-Feb-2015.) |
Ref | Expression |
---|---|
0we1 | ⊢ ∅ We 1o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | br0 5191 | . . 3 ⊢ ¬ ∅∅∅ | |
2 | rel0 5792 | . . . 4 ⊢ Rel ∅ | |
3 | wesn 5757 | . . . 4 ⊢ (Rel ∅ → (∅ We {∅} ↔ ¬ ∅∅∅)) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (∅ We {∅} ↔ ¬ ∅∅∅) |
5 | 1, 4 | mpbir 230 | . 2 ⊢ ∅ We {∅} |
6 | df1o2 8457 | . . 3 ⊢ 1o = {∅} | |
7 | weeq2 5659 | . . 3 ⊢ (1o = {∅} → (∅ We 1o ↔ ∅ We {∅})) | |
8 | 6, 7 | ax-mp 5 | . 2 ⊢ (∅ We 1o ↔ ∅ We {∅}) |
9 | 5, 8 | mpbir 230 | 1 ⊢ ∅ We 1o |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1541 ∅c0 4319 {csn 4623 class class class wbr 5142 We wwe 5624 Rel wrel 5675 1oc1o 8443 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pr 5421 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5143 df-opab 5205 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-suc 6360 df-1o 8450 |
This theorem is referenced by: psr1tos 21644 |
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