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Theorem onunel 33433
Description: The union of two ordinals is in a third iff both of the first two are. (Contributed by Scott Fenton, 10-Sep-2024.)
Assertion
Ref Expression
onunel ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝐵) ∈ 𝐶 ↔ (𝐴𝐶𝐵𝐶)))

Proof of Theorem onunel
StepHypRef Expression
1 ssequn1 4110 . . . . . 6 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)
21biimpi 219 . . . . 5 (𝐴𝐵 → (𝐴𝐵) = 𝐵)
32eleq1d 2824 . . . 4 (𝐴𝐵 → ((𝐴𝐵) ∈ 𝐶𝐵𝐶))
43adantl 485 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → ((𝐴𝐵) ∈ 𝐶𝐵𝐶))
5 ontr2 6280 . . . . . 6 ((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
653adant2 1133 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
76expdimp 456 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → (𝐵𝐶𝐴𝐶))
87pm4.71rd 566 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → (𝐵𝐶 ↔ (𝐴𝐶𝐵𝐶)))
94, 8bitrd 282 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴𝐵) → ((𝐴𝐵) ∈ 𝐶 ↔ (𝐴𝐶𝐵𝐶)))
10 ssequn2 4113 . . . . . 6 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐴)
1110biimpi 219 . . . . 5 (𝐵𝐴 → (𝐴𝐵) = 𝐴)
1211eleq1d 2824 . . . 4 (𝐵𝐴 → ((𝐴𝐵) ∈ 𝐶𝐴𝐶))
1312adantl 485 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵𝐴) → ((𝐴𝐵) ∈ 𝐶𝐴𝐶))
14 ontr2 6280 . . . . . 6 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵𝐴𝐴𝐶) → 𝐵𝐶))
15143adant1 1132 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵𝐴𝐴𝐶) → 𝐵𝐶))
1615expdimp 456 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵𝐴) → (𝐴𝐶𝐵𝐶))
1716pm4.71d 565 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵𝐴) → (𝐴𝐶 ↔ (𝐴𝐶𝐵𝐶)))
1813, 17bitrd 282 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐵𝐴) → ((𝐴𝐵) ∈ 𝐶 ↔ (𝐴𝐶𝐵𝐶)))
19 eloni 6243 . . . 4 (𝐴 ∈ On → Ord 𝐴)
20 eloni 6243 . . . 4 (𝐵 ∈ On → Ord 𝐵)
21 ordtri2or2 6329 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))
2219, 20, 21syl2an 599 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵𝐵𝐴))
23223adant3 1134 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵𝐵𝐴))
249, 18, 23mpjaodan 959 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝐵) ∈ 𝐶 ↔ (𝐴𝐶𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 847  w3a 1089   = wceq 1543  wcel 2112  cun 3880  wss 3882  Ord word 6232  Oncon0 6233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-11 2160  ax-ext 2710  ax-sep 5208  ax-nul 5215  ax-pr 5338
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-pss 3901  df-nul 4254  df-if 4456  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4836  df-br 5070  df-opab 5132  df-tr 5178  df-eprel 5477  df-po 5485  df-so 5486  df-fr 5526  df-we 5528  df-ord 6236  df-on 6237
This theorem is referenced by: (None)
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