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Theorem ordunidif 6012
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 5988 . . . . . . . 8 ((Ord 𝐴𝐵𝐴) → 𝐵 ∈ On)
2 onelss 6006 . . . . . . . 8 (𝐵 ∈ On → (𝑥𝐵𝑥𝐵))
31, 2syl 17 . . . . . . 7 ((Ord 𝐴𝐵𝐴) → (𝑥𝐵𝑥𝐵))
4 eloni 5974 . . . . . . . . . . 11 (𝐵 ∈ On → Ord 𝐵)
5 ordirr 5982 . . . . . . . . . . 11 (Ord 𝐵 → ¬ 𝐵𝐵)
64, 5syl 17 . . . . . . . . . 10 (𝐵 ∈ On → ¬ 𝐵𝐵)
7 eldif 3809 . . . . . . . . . . 11 (𝐵 ∈ (𝐴𝐵) ↔ (𝐵𝐴 ∧ ¬ 𝐵𝐵))
87simplbi2 496 . . . . . . . . . 10 (𝐵𝐴 → (¬ 𝐵𝐵𝐵 ∈ (𝐴𝐵)))
96, 8syl5 34 . . . . . . . . 9 (𝐵𝐴 → (𝐵 ∈ On → 𝐵 ∈ (𝐴𝐵)))
109adantl 475 . . . . . . . 8 ((Ord 𝐴𝐵𝐴) → (𝐵 ∈ On → 𝐵 ∈ (𝐴𝐵)))
111, 10mpd 15 . . . . . . 7 ((Ord 𝐴𝐵𝐴) → 𝐵 ∈ (𝐴𝐵))
123, 11jctild 523 . . . . . 6 ((Ord 𝐴𝐵𝐴) → (𝑥𝐵 → (𝐵 ∈ (𝐴𝐵) ∧ 𝑥𝐵)))
1312adantr 474 . . . . 5 (((Ord 𝐴𝐵𝐴) ∧ 𝑥𝐴) → (𝑥𝐵 → (𝐵 ∈ (𝐴𝐵) ∧ 𝑥𝐵)))
14 sseq2 3853 . . . . . 6 (𝑦 = 𝐵 → (𝑥𝑦𝑥𝐵))
1514rspcev 3527 . . . . 5 ((𝐵 ∈ (𝐴𝐵) ∧ 𝑥𝐵) → ∃𝑦 ∈ (𝐴𝐵)𝑥𝑦)
1613, 15syl6 35 . . . 4 (((Ord 𝐴𝐵𝐴) ∧ 𝑥𝐴) → (𝑥𝐵 → ∃𝑦 ∈ (𝐴𝐵)𝑥𝑦))
17 eldif 3809 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
1817biimpri 220 . . . . . . . 8 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥 ∈ (𝐴𝐵))
19 ssid 3849 . . . . . . . 8 𝑥𝑥
2018, 19jctir 518 . . . . . . 7 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝑥))
2120ex 403 . . . . . 6 (𝑥𝐴 → (¬ 𝑥𝐵 → (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝑥)))
22 sseq2 3853 . . . . . . 7 (𝑦 = 𝑥 → (𝑥𝑦𝑥𝑥))
2322rspcev 3527 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝑥) → ∃𝑦 ∈ (𝐴𝐵)𝑥𝑦)
2421, 23syl6 35 . . . . 5 (𝑥𝐴 → (¬ 𝑥𝐵 → ∃𝑦 ∈ (𝐴𝐵)𝑥𝑦))
2524adantl 475 . . . 4 (((Ord 𝐴𝐵𝐴) ∧ 𝑥𝐴) → (¬ 𝑥𝐵 → ∃𝑦 ∈ (𝐴𝐵)𝑥𝑦))
2616, 25pm2.61d 172 . . 3 (((Ord 𝐴𝐵𝐴) ∧ 𝑥𝐴) → ∃𝑦 ∈ (𝐴𝐵)𝑥𝑦)
2726ralrimiva 3176 . 2 ((Ord 𝐴𝐵𝐴) → ∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝑥𝑦)
28 unidif 4694 . 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 386   = wceq 1658  wcel 2166  wral 3118  wrex 3119  cdif 3796  wss 3799   cuni 4659  Ord word 5963  Oncon0 5964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-tr 4977  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-ord 5967  df-on 5968
This theorem is referenced by: (None)
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