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Theorem onsupnmax 43240
Description: If the union of a class of ordinals is not the maximum element of that class, then the union is a limit ordinal or empty. But this isn't a biconditional since 𝐴 could be a non-empty set where a limit ordinal or the empty set happens to be the largest element. (Contributed by RP, 27-Jan-2025.)
Assertion
Ref Expression
onsupnmax (𝐴 ⊆ On → (¬ 𝐴𝐴 𝐴 = 𝐴))

Proof of Theorem onsupnmax
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexnal 3100 . . . . . . . . 9 (∃𝑥𝐴 ¬ ∃𝑦𝐴 𝑥𝑦 ↔ ¬ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)
2 ralnex 3072 . . . . . . . . . 10 (∀𝑦𝐴 ¬ 𝑥𝑦 ↔ ¬ ∃𝑦𝐴 𝑥𝑦)
32rexbii 3094 . . . . . . . . 9 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦 ↔ ∃𝑥𝐴 ¬ ∃𝑦𝐴 𝑥𝑦)
4 ssunib 43232 . . . . . . . . . 10 (𝐴 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)
54notbii 320 . . . . . . . . 9 𝐴 𝐴 ↔ ¬ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)
61, 3, 53bitr4ri 304 . . . . . . . 8 𝐴 𝐴 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
7 simpll 767 . . . . . . . . . . . 12 (((𝐴 ⊆ On ∧ 𝐴 ∈ On) ∧ 𝑥𝐴) → 𝐴 ⊆ On)
87sselda 3983 . . . . . . . . . . 11 ((((𝐴 ⊆ On ∧ 𝐴 ∈ On) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → 𝑦 ∈ On)
9 simpl 482 . . . . . . . . . . . . 13 ((𝐴 ⊆ On ∧ 𝐴 ∈ On) → 𝐴 ⊆ On)
109sselda 3983 . . . . . . . . . . . 12 (((𝐴 ⊆ On ∧ 𝐴 ∈ On) ∧ 𝑥𝐴) → 𝑥 ∈ On)
1110adantr 480 . . . . . . . . . . 11 ((((𝐴 ⊆ On ∧ 𝐴 ∈ On) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → 𝑥 ∈ On)
12 ontri1 6418 . . . . . . . . . . 11 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦𝑥 ↔ ¬ 𝑥𝑦))
138, 11, 12syl2anc 584 . . . . . . . . . 10 ((((𝐴 ⊆ On ∧ 𝐴 ∈ On) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → (𝑦𝑥 ↔ ¬ 𝑥𝑦))
1413ralbidva 3176 . . . . . . . . 9 (((𝐴 ⊆ On ∧ 𝐴 ∈ On) ∧ 𝑥𝐴) → (∀𝑦𝐴 𝑦𝑥 ↔ ∀𝑦𝐴 ¬ 𝑥𝑦))
1514rexbidva 3177 . . . . . . . 8 ((𝐴 ⊆ On ∧ 𝐴 ∈ On) → (∃𝑥𝐴𝑦𝐴 𝑦𝑥 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦))
166, 15bitr4id 290 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴 ∈ On) → (¬ 𝐴 𝐴 ↔ ∃𝑥𝐴𝑦𝐴 𝑦𝑥))
17 unielid 43231 . . . . . . . . 9 ( 𝐴𝐴 ↔ ∃𝑥𝐴𝑦𝐴 𝑦𝑥)
1817a1i 11 . . . . . . . 8 ((𝐴 ⊆ On ∧ 𝐴 ∈ On) → ( 𝐴𝐴 ↔ ∃𝑥𝐴𝑦𝐴 𝑦𝑥))
1918biimprd 248 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴 ∈ On) → (∃𝑥𝐴𝑦𝐴 𝑦𝑥 𝐴𝐴))
2016, 19sylbid 240 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴 ∈ On) → (¬ 𝐴 𝐴 𝐴𝐴))
2120con1d 145 . . . . 5 ((𝐴 ⊆ On ∧ 𝐴 ∈ On) → (¬ 𝐴𝐴𝐴 𝐴))
22 uniss 4915 . . . . 5 (𝐴 𝐴 𝐴 𝐴)
2321, 22syl6 35 . . . 4 ((𝐴 ⊆ On ∧ 𝐴 ∈ On) → (¬ 𝐴𝐴 𝐴 𝐴))
24 ssorduni 7799 . . . . . . . 8 (𝐴 ⊆ On → Ord 𝐴)
25 orduniss 6481 . . . . . . . 8 (Ord 𝐴 𝐴 𝐴)
2624, 25syl 17 . . . . . . 7 (𝐴 ⊆ On → 𝐴 𝐴)
2726biantrud 531 . . . . . 6 (𝐴 ⊆ On → ( 𝐴 𝐴 ↔ ( 𝐴 𝐴 𝐴 𝐴)))
28 eqss 3999 . . . . . 6 ( 𝐴 = 𝐴 ↔ ( 𝐴 𝐴 𝐴 𝐴))
2927, 28bitr4di 289 . . . . 5 (𝐴 ⊆ On → ( 𝐴 𝐴 𝐴 = 𝐴))
3029adantr 480 . . . 4 ((𝐴 ⊆ On ∧ 𝐴 ∈ On) → ( 𝐴 𝐴 𝐴 = 𝐴))
3123, 30sylibd 239 . . 3 ((𝐴 ⊆ On ∧ 𝐴 ∈ On) → (¬ 𝐴𝐴 𝐴 = 𝐴))
3231ex 412 . 2 (𝐴 ⊆ On → ( 𝐴 ∈ On → (¬ 𝐴𝐴 𝐴 = 𝐴)))
33 unon 7851 . . . . 5 On = On
3433a1i 11 . . . 4 ( 𝐴 = On → On = On)
35 unieq 4918 . . . 4 ( 𝐴 = On → 𝐴 = On)
36 id 22 . . . 4 ( 𝐴 = On → 𝐴 = On)
3734, 35, 363eqtr4rd 2788 . . 3 ( 𝐴 = On → 𝐴 = 𝐴)
3837a1i13 27 . 2 (𝐴 ⊆ On → ( 𝐴 = On → (¬ 𝐴𝐴 𝐴 = 𝐴)))
39 ordeleqon 7802 . . 3 (Ord 𝐴 ↔ ( 𝐴 ∈ On ∨ 𝐴 = On))
4024, 39sylib 218 . 2 (𝐴 ⊆ On → ( 𝐴 ∈ On ∨ 𝐴 = On))
4132, 38, 40mpjaod 861 1 (𝐴 ⊆ On → (¬ 𝐴𝐴 𝐴 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wral 3061  wrex 3070  wss 3951   cuni 4907  Ord word 6383  Oncon0 6384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-suc 6390
This theorem is referenced by:  onsupeqnmax  43259  onsupsucismax  43292
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