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Theorem pm110.643ALT 9081
Description: Alternate proof of pm110.643 9080. (Contributed by Mario Carneiro, 29-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
pm110.643ALT (1𝑜 +𝑐 1𝑜) ≈ 2𝑜

Proof of Theorem pm110.643ALT
StepHypRef Expression
1 1on 7655 . . 3 1𝑜 ∈ On
21onordi 5913 . . . 4 Ord 1𝑜
3 ordirr 5822 . . . 4 (Ord 1𝑜 → ¬ 1𝑜 ∈ 1𝑜)
42, 3ax-mp 5 . . 3 ¬ 1𝑜 ∈ 1𝑜
5 cda1en 9078 . . 3 ((1𝑜 ∈ On ∧ ¬ 1𝑜 ∈ 1𝑜) → (1𝑜 +𝑐 1𝑜) ≈ suc 1𝑜)
61, 4, 5mp2an 710 . 2 (1𝑜 +𝑐 1𝑜) ≈ suc 1𝑜
7 df-2o 7649 . 2 2𝑜 = suc 1𝑜
86, 7breqtrri 4755 1 (1𝑜 +𝑐 1𝑜) ≈ 2𝑜
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2071   class class class wbr 4728  Ord word 5803  Oncon0 5804  suc csuc 5806  (class class class)co 6733  1𝑜c1o 7641  2𝑜c2o 7642  cen 8037   +𝑐 ccda 9070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-8 2073  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672  ax-sep 4857  ax-nul 4865  ax-pow 4916  ax-pr 4979  ax-un 7034
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-eu 2543  df-mo 2544  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ne 2865  df-ral 2987  df-rex 2988  df-rab 2991  df-v 3274  df-sbc 3510  df-dif 3651  df-un 3653  df-in 3655  df-ss 3662  df-pss 3664  df-nul 3992  df-if 4163  df-pw 4236  df-sn 4254  df-pr 4256  df-tp 4258  df-op 4260  df-uni 4513  df-int 4552  df-br 4729  df-opab 4789  df-mpt 4806  df-tr 4829  df-id 5096  df-eprel 5101  df-po 5107  df-so 5108  df-fr 5145  df-we 5147  df-xp 5192  df-rel 5193  df-cnv 5194  df-co 5195  df-dm 5196  df-rn 5197  df-res 5198  df-ima 5199  df-ord 5807  df-on 5808  df-suc 5810  df-iota 5932  df-fun 5971  df-fn 5972  df-f 5973  df-f1 5974  df-fo 5975  df-f1o 5976  df-fv 5977  df-ov 6736  df-oprab 6737  df-mpt2 6738  df-1o 7648  df-2o 7649  df-er 7830  df-en 8041  df-cda 9071
This theorem is referenced by: (None)
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