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Theorem dju1p1e2ALT 9451
Description: Alternate proof of dju1p1e2 9450. (Contributed by Mario Carneiro, 29-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dju1p1e2ALT (1o ⊔ 1o) ≈ 2o

Proof of Theorem dju1p1e2ALT
StepHypRef Expression
1 1on 7965 . . 3 1o ∈ On
21onordi 6175 . . . 4 Ord 1o
3 ordirr 6089 . . . 4 (Ord 1o → ¬ 1o ∈ 1o)
42, 3ax-mp 5 . . 3 ¬ 1o ∈ 1o
5 dju1en 9448 . . 3 ((1o ∈ On ∧ ¬ 1o ∈ 1o) → (1o ⊔ 1o) ≈ suc 1o)
61, 4, 5mp2an 688 . 2 (1o ⊔ 1o) ≈ suc 1o
7 df-2o 7959 . 2 2o = suc 1o
86, 7breqtrri 4993 1 (1o ⊔ 1o) ≈ 2o
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2081   class class class wbr 4966  Ord word 6070  Oncon0 6071  suc csuc 6073  1oc1o 7951  2oc2o 7952  cen 8359  cdju 9178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3710  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-pss 3880  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-tp 4481  df-op 4483  df-uni 4750  df-int 4787  df-br 4967  df-opab 5029  df-mpt 5046  df-tr 5069  df-id 5353  df-eprel 5358  df-po 5367  df-so 5368  df-fr 5407  df-we 5409  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-ord 6074  df-on 6075  df-suc 6077  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-1st 7550  df-2nd 7551  df-1o 7958  df-2o 7959  df-er 8144  df-en 8363  df-dju 9181
This theorem is referenced by: (None)
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