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Theorem ordunidif 6237
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 6213 . . . . . . . 8 ((Ord 𝐴𝐵𝐴) → 𝐵 ∈ On)
2 onelss 6231 . . . . . . . 8 (𝐵 ∈ On → (𝑥𝐵𝑥𝐵))
31, 2syl 17 . . . . . . 7 ((Ord 𝐴𝐵𝐴) → (𝑥𝐵𝑥𝐵))
4 eloni 6199 . . . . . . . . . . 11 (𝐵 ∈ On → Ord 𝐵)
5 ordirr 6207 . . . . . . . . . . 11 (Ord 𝐵 → ¬ 𝐵𝐵)
64, 5syl 17 . . . . . . . . . 10 (𝐵 ∈ On → ¬ 𝐵𝐵)
7 eldif 3950 . . . . . . . . . . 11 (𝐵 ∈ (𝐴𝐵) ↔ (𝐵𝐴 ∧ ¬ 𝐵𝐵))
87simplbi2 501 . . . . . . . . . 10 (𝐵𝐴 → (¬ 𝐵𝐵𝐵 ∈ (𝐴𝐵)))
96, 8syl5 34 . . . . . . . . 9 (𝐵𝐴 → (𝐵 ∈ On → 𝐵 ∈ (𝐴𝐵)))
109adantl 482 . . . . . . . 8 ((Ord 𝐴𝐵𝐴) → (𝐵 ∈ On → 𝐵 ∈ (𝐴𝐵)))
111, 10mpd 15 . . . . . . 7 ((Ord 𝐴𝐵𝐴) → 𝐵 ∈ (𝐴𝐵))
123, 11jctild 526 . . . . . 6 ((Ord 𝐴𝐵𝐴) → (𝑥𝐵 → (𝐵 ∈ (𝐴𝐵) ∧ 𝑥𝐵)))
1312adantr 481 . . . . 5 (((Ord 𝐴𝐵𝐴) ∧ 𝑥𝐴) → (𝑥𝐵 → (𝐵 ∈ (𝐴𝐵) ∧ 𝑥𝐵)))
14 sseq2 3997 . . . . . 6 (𝑦 = 𝐵 → (𝑥𝑦𝑥𝐵))
1514rspcev 3627 . . . . 5 ((𝐵 ∈ (𝐴𝐵) ∧ 𝑥𝐵) → ∃𝑦 ∈ (𝐴𝐵)𝑥𝑦)
1613, 15syl6 35 . . . 4 (((Ord 𝐴𝐵𝐴) ∧ 𝑥𝐴) → (𝑥𝐵 → ∃𝑦 ∈ (𝐴𝐵)𝑥𝑦))
17 eldif 3950 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
1817biimpri 229 . . . . . . . 8 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → 𝑥 ∈ (𝐴𝐵))
19 ssid 3993 . . . . . . . 8 𝑥𝑥
2018, 19jctir 521 . . . . . . 7 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝑥))
2120ex 413 . . . . . 6 (𝑥𝐴 → (¬ 𝑥𝐵 → (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝑥)))
22 sseq2 3997 . . . . . . 7 (𝑦 = 𝑥 → (𝑥𝑦𝑥𝑥))
2322rspcev 3627 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝑥) → ∃𝑦 ∈ (𝐴𝐵)𝑥𝑦)
2421, 23syl6 35 . . . . 5 (𝑥𝐴 → (¬ 𝑥𝐵 → ∃𝑦 ∈ (𝐴𝐵)𝑥𝑦))
2524adantl 482 . . . 4 (((Ord 𝐴𝐵𝐴) ∧ 𝑥𝐴) → (¬ 𝑥𝐵 → ∃𝑦 ∈ (𝐴𝐵)𝑥𝑦))
2616, 25pm2.61d 180 . . 3 (((Ord 𝐴𝐵𝐴) ∧ 𝑥𝐴) → ∃𝑦 ∈ (𝐴𝐵)𝑥𝑦)
2726ralrimiva 3187 . 2 ((Ord 𝐴𝐵𝐴) → ∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝑥𝑦)
28 unidif 4870 . 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 396   = wceq 1530  wcel 2107  wral 3143  wrex 3144  cdif 3937  wss 3940   cuni 4837  Ord word 6188  Oncon0 6189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-tr 5170  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-ord 6192  df-on 6193
This theorem is referenced by: (None)
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