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Mirrors > Home > MPE Home > Th. List > df2o3 | Structured version Visualization version GIF version |
Description: Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
df2o3 | ⊢ 2𝑜 = {∅, 1𝑜} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2o 7448 | . 2 ⊢ 2𝑜 = suc 1𝑜 | |
2 | df-suc 5646 | . 2 ⊢ suc 1𝑜 = (1𝑜 ∪ {1𝑜}) | |
3 | df1o2 7459 | . . . 4 ⊢ 1𝑜 = {∅} | |
4 | 3 | uneq1i 3725 | . . 3 ⊢ (1𝑜 ∪ {1𝑜}) = ({∅} ∪ {1𝑜}) |
5 | df-pr 4128 | . . 3 ⊢ {∅, 1𝑜} = ({∅} ∪ {1𝑜}) | |
6 | 4, 5 | eqtr4i 2635 | . 2 ⊢ (1𝑜 ∪ {1𝑜}) = {∅, 1𝑜} |
7 | 1, 2, 6 | 3eqtri 2636 | 1 ⊢ 2𝑜 = {∅, 1𝑜} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∪ cun 3538 ∅c0 3874 {csn 4125 {cpr 4127 suc csuc 5642 1𝑜c1o 7440 2𝑜c2o 7441 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-v 3175 df-dif 3543 df-un 3545 df-nul 3875 df-pr 4128 df-suc 5646 df-1o 7447 df-2o 7448 |
This theorem is referenced by: df2o2 7461 2oconcl 7470 map2xp 8015 1sdom 8048 cantnflem2 8470 xp2cda 8885 sdom2en01 9007 sadcf 15013 xpscfn 16042 xpscfv 16045 xpsfrnel 16046 xpsfeq 16047 xpsfrnel2 16048 xpsle 16064 setcepi 16561 efgi0 17956 efgi1 17957 vrgpf 18004 vrgpinv 18005 frgpuptinv 18007 frgpup2 18012 frgpup3lem 18013 frgpnabllem1 18099 dmdprdpr 18271 dprdpr 18272 xpstopnlem1 21422 xpstopnlem2 21424 xpsxmetlem 21994 xpsdsval 21996 xpsmet 21997 onint1 31618 pw2f1ocnv 36622 wepwsolem 36630 df3o2 37342 clsk1independent 37364 |
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