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Theorem ordunidif 6222
 Description: The union of an ordinal stays the same if a subset equal to one of its elements is removed. (Contributed by NM, 10-Dec-2004.)
Assertion
Ref Expression
ordunidif ((Ord 𝐴𝐵𝐴) → (𝐴𝐵) = 𝐴)

Proof of Theorem ordunidif
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordelon 6198 . . . . . . . 8 ((Ord 𝐴𝐵𝐴) → 𝐵 ∈ On)
2 onelss 6216 . . . . . . . 8 (𝐵 ∈ On → (𝑥𝐵𝑥𝐵))
31, 2syl 17 . . . . . . 7 ((Ord 𝐴𝐵𝐴) → (𝑥𝐵𝑥𝐵))
4 eloni 6184 . . . . . . . . . . 11 (𝐵 ∈ On → Ord 𝐵)
5 ordirr 6192 . . . . . . . . . . 11 (Ord 𝐵 → ¬ 𝐵𝐵)
64, 5syl 17 . . . . . . . . . 10 (𝐵 ∈ On → ¬ 𝐵𝐵)
7 eldif 3870 . . . . . . . . . . 11 (𝐵 ∈ (𝐴𝐵) ↔ (𝐵𝐴 ∧ ¬ 𝐵𝐵))
87simplbi2 504 . . . . . . . . . 10 (𝐵𝐴 → (¬ 𝐵𝐵𝐵 ∈ (𝐴𝐵)))
96, 8syl5 34 . . . . . . . . 9 (𝐵𝐴 → (𝐵 ∈ On → 𝐵 ∈ (𝐴𝐵)))
109adantl 485 . . . . . . . 8 ((Ord 𝐴𝐵𝐴) → (𝐵 ∈ On → 𝐵 ∈ (𝐴𝐵)))
111, 10mpd 15 . . . . . . 7 ((Ord 𝐴𝐵𝐴) → 𝐵 ∈ (𝐴𝐵))
123, 11jctild 529 . . . . . 6 ((Ord 𝐴𝐵𝐴) → (𝑥𝐵 → (𝐵 ∈ (𝐴𝐵) ∧ 𝑥𝐵)))
1312adantr 484 . . . . 5 (((Ord 𝐴𝐵𝐴) ∧ 𝑥𝐴) → (𝑥𝐵 → (𝐵 ∈ (𝐴𝐵) ∧ 𝑥𝐵)))
14 sseq2 3920 . . . . . 6 (𝑦 = 𝐵 → (𝑥𝑦𝑥𝐵))
1514rspcev 3543 . . . . 5 ((𝐵 ∈ (𝐴𝐵) ∧ 𝑥𝐵) → ∃𝑦 ∈ (𝐴𝐵)𝑥𝑦)
1613, 15syl6 35 . . . 4 (((Ord 𝐴𝐵𝐴) ∧ 𝑥𝐴) → (𝑥𝐵 → ∃𝑦 ∈ (𝐴𝐵)𝑥𝑦))
17 eldif 3870 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
1817biimpri 231 . . . . . . . 8 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥 ∈ (𝐴𝐵))
19 ssid 3916 . . . . . . . 8 𝑥𝑥
2018, 19jctir 524 . . . . . . 7 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝑥))
2120ex 416 . . . . . 6 (𝑥𝐴 → (¬ 𝑥𝐵 → (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝑥)))
22 sseq2 3920 . . . . . . 7 (𝑦 = 𝑥 → (𝑥𝑦𝑥𝑥))
2322rspcev 3543 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝑥) → ∃𝑦 ∈ (𝐴𝐵)𝑥𝑦)
2421, 23syl6 35 . . . . 5 (𝑥𝐴 → (¬ 𝑥𝐵 → ∃𝑦 ∈ (𝐴𝐵)𝑥𝑦))
2524adantl 485 . . . 4 (((Ord 𝐴𝐵𝐴) ∧ 𝑥𝐴) → (¬ 𝑥𝐵 → ∃𝑦 ∈ (𝐴𝐵)𝑥𝑦))
2616, 25pm2.61d 182 . . 3 (((Ord 𝐴𝐵𝐴) ∧ 𝑥𝐴) → ∃𝑦 ∈ (𝐴𝐵)𝑥𝑦)
2726ralrimiva 3113 . 2 ((Ord 𝐴𝐵𝐴) → ∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝑥𝑦)
28 unidif 4837 . 2 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝑥𝑦 (𝐴𝐵) = 𝐴)
2927, 28syl 17 1 ((Ord 𝐴𝐵𝐴) → (𝐴𝐵) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071   ∖ cdif 3857   ⊆ wss 3860  ∪ cuni 4801  Ord word 6173  Oncon0 6174 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-tr 5143  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-ord 6177  df-on 6178 This theorem is referenced by: (None)
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