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Mirrors > Home > MPE Home > Th. List > dju1p1e2ALT | Structured version Visualization version GIF version |
Description: Alternate proof of dju1p1e2 9450. (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 7965 | . . 3 ⊢ 1o ∈ On | |
2 | 1 | onordi 6175 | . . . 4 ⊢ Ord 1o |
3 | ordirr 6089 | . . . 4 ⊢ (Ord 1o → ¬ 1o ∈ 1o) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ ¬ 1o ∈ 1o |
5 | dju1en 9448 | . . 3 ⊢ ((1o ∈ On ∧ ¬ 1o ∈ 1o) → (1o ⊔ 1o) ≈ suc 1o) | |
6 | 1, 4, 5 | mp2an 688 | . 2 ⊢ (1o ⊔ 1o) ≈ suc 1o |
7 | df-2o 7959 | . 2 ⊢ 2o = suc 1o | |
8 | 6, 7 | breqtrri 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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