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Mirrors > Home > MPE Home > Th. List > dju1p1e2ALT | Structured version Visualization version GIF version |
Description: Alternate proof of dju1p1e2 10067. (Contributed by Mario Carneiro, 29-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dju1p1e2ALT | ⊢ (1o ⊔ 1o) ≈ 2o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1on 8416 | . . 3 ⊢ 1o ∈ On | |
2 | 1 | onordi 6425 | . . . 4 ⊢ Ord 1o |
3 | ordirr 6333 | . . . 4 ⊢ (Ord 1o → ¬ 1o ∈ 1o) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ ¬ 1o ∈ 1o |
5 | dju1en 10065 | . . 3 ⊢ ((1o ∈ On ∧ ¬ 1o ∈ 1o) → (1o ⊔ 1o) ≈ suc 1o) | |
6 | 1, 4, 5 | mp2an 690 | . 2 ⊢ (1o ⊔ 1o) ≈ suc 1o |
7 | df-2o 8405 | . 2 ⊢ 2o = suc 1o | |
8 | 6, 7 | breqtrri 5130 | 1 ⊢ (1o ⊔ 1o) ≈ 2o |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2106 class class class wbr 5103 Ord word 6314 Oncon0 6315 suc csuc 6317 1oc1o 8397 2oc2o 8398 ≈ cen 8838 ⊔ cdju 9792 |
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-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
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-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ord 6318 df-on 6319 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-1st 7913 df-2nd 7914 df-1o 8404 df-2o 8405 df-er 8606 df-en 8842 df-dju 9795 |
This theorem is referenced by: (None) |
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