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Theorem ordunidif 6434
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 6409 . . . . . . . 8 ((Ord 𝐴𝐵𝐴) → 𝐵 ∈ On)
2 onelss 6427 . . . . . . . 8 (𝐵 ∈ On → (𝑥𝐵𝑥𝐵))
31, 2syl 17 . . . . . . 7 ((Ord 𝐴𝐵𝐴) → (𝑥𝐵𝑥𝐵))
4 eloni 6395 . . . . . . . . . . 11 (𝐵 ∈ On → Ord 𝐵)
5 ordirr 6403 . . . . . . . . . . 11 (Ord 𝐵 → ¬ 𝐵𝐵)
64, 5syl 17 . . . . . . . . . 10 (𝐵 ∈ On → ¬ 𝐵𝐵)
7 eldif 3972 . . . . . . . . . . 11 (𝐵 ∈ (𝐴𝐵) ↔ (𝐵𝐴 ∧ ¬ 𝐵𝐵))
87simplbi2 500 . . . . . . . . . 10 (𝐵𝐴 → (¬ 𝐵𝐵𝐵 ∈ (𝐴𝐵)))
96, 8syl5 34 . . . . . . . . 9 (𝐵𝐴 → (𝐵 ∈ On → 𝐵 ∈ (𝐴𝐵)))
109adantl 481 . . . . . . . 8 ((Ord 𝐴𝐵𝐴) → (𝐵 ∈ On → 𝐵 ∈ (𝐴𝐵)))
111, 10mpd 15 . . . . . . 7 ((Ord 𝐴𝐵𝐴) → 𝐵 ∈ (𝐴𝐵))
123, 11jctild 525 . . . . . 6 ((Ord 𝐴𝐵𝐴) → (𝑥𝐵 → (𝐵 ∈ (𝐴𝐵) ∧ 𝑥𝐵)))
1312adantr 480 . . . . 5 (((Ord 𝐴𝐵𝐴) ∧ 𝑥𝐴) → (𝑥𝐵 → (𝐵 ∈ (𝐴𝐵) ∧ 𝑥𝐵)))
14 sseq2 4021 . . . . . 6 (𝑦 = 𝐵 → (𝑥𝑦𝑥𝐵))
1514rspcev 3621 . . . . 5 ((𝐵 ∈ (𝐴𝐵) ∧ 𝑥𝐵) → ∃𝑦 ∈ (𝐴𝐵)𝑥𝑦)
1613, 15syl6 35 . . . 4 (((Ord 𝐴𝐵𝐴) ∧ 𝑥𝐴) → (𝑥𝐵 → ∃𝑦 ∈ (𝐴𝐵)𝑥𝑦))
17 eldif 3972 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
1817biimpri 228 . . . . . . . 8 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥 ∈ (𝐴𝐵))
19 ssid 4017 . . . . . . . 8 𝑥𝑥
2018, 19jctir 520 . . . . . . 7 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝑥))
2120ex 412 . . . . . 6 (𝑥𝐴 → (¬ 𝑥𝐵 → (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝑥)))
22 sseq2 4021 . . . . . . 7 (𝑦 = 𝑥 → (𝑥𝑦𝑥𝑥))
2322rspcev 3621 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝑥) → ∃𝑦 ∈ (𝐴𝐵)𝑥𝑦)
2421, 23syl6 35 . . . . 5 (𝑥𝐴 → (¬ 𝑥𝐵 → ∃𝑦 ∈ (𝐴𝐵)𝑥𝑦))
2524adantl 481 . . . 4 (((Ord 𝐴𝐵𝐴) ∧ 𝑥𝐴) → (¬ 𝑥𝐵 → ∃𝑦 ∈ (𝐴𝐵)𝑥𝑦))
2616, 25pm2.61d 179 . . 3 (((Ord 𝐴𝐵𝐴) ∧ 𝑥𝐴) → ∃𝑦 ∈ (𝐴𝐵)𝑥𝑦)
2726ralrimiva 3143 . 2 ((Ord 𝐴𝐵𝐴) → ∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝑥𝑦)
28 unidif 4946 . 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 395   = wceq 1536  wcel 2105  wral 3058  wrex 3067  cdif 3959  wss 3962   cuni 4911  Ord word 6384  Oncon0 6385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-ord 6388  df-on 6389
This theorem is referenced by: (None)
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