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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 4631 | . . 3 ⊢ ¬ ∅∅∅ | |
2 | rel0 5166 | . . . 4 ⊢ Rel ∅ | |
3 | wesn 5113 | . . . 4 ⊢ (Rel ∅ → (∅ We {∅} ↔ ¬ ∅∅∅)) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (∅ We {∅} ↔ ¬ ∅∅∅) |
5 | 1, 4 | mpbir 220 | . 2 ⊢ ∅ We {∅} |
6 | df1o2 7459 | . . 3 ⊢ 1𝑜 = {∅} | |
7 | weeq2 5027 | . . 3 ⊢ (1𝑜 = {∅} → (∅ We 1𝑜 ↔ ∅ We {∅})) | |
8 | 6, 7 | ax-mp 5 | . 2 ⊢ (∅ We 1𝑜 ↔ ∅ We {∅}) |
9 | 5, 8 | mpbir 220 | 1 ⊢ ∅ We 1𝑜 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 = wceq 1475 ∅c0 3874 {csn 4125 class class class wbr 4583 We wwe 4996 Rel wrel 5043 1𝑜c1o 7440 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-suc 5646 df-1o 7447 |
This theorem is referenced by: psr1tos 19380 |
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