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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 5099 | . . 3 ⊢ ¬ ∅∅∅ | |
2 | rel0 5666 | . . . 4 ⊢ Rel ∅ | |
3 | wesn 5634 | . . . 4 ⊢ (Rel ∅ → (∅ We {∅} ↔ ¬ ∅∅∅)) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (∅ We {∅} ↔ ¬ ∅∅∅) |
5 | 1, 4 | mpbir 234 | . 2 ⊢ ∅ We {∅} |
6 | df1o2 8211 | . . 3 ⊢ 1o = {∅} | |
7 | weeq2 5537 | . . 3 ⊢ (1o = {∅} → (∅ We 1o ↔ ∅ We {∅})) | |
8 | 6, 7 | ax-mp 5 | . 2 ⊢ (∅ We 1o ↔ ∅ We {∅}) |
9 | 5, 8 | mpbir 234 | 1 ⊢ ∅ We 1o |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 = wceq 1543 ∅c0 4234 {csn 4538 class class class wbr 5050 We wwe 5505 Rel wrel 5553 1oc1o 8192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pr 5319 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2940 df-ral 3063 df-rex 3064 df-rab 3067 df-v 3407 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-nul 4235 df-if 4437 df-sn 4539 df-pr 4541 df-op 4545 df-br 5051 df-opab 5113 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-xp 5554 df-rel 5555 df-suc 6216 df-1o 8199 |
This theorem is referenced by: psr1tos 21107 |
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