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Theorem onsupnmax 42721
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 3090 . . . . . . . . 9 (∃𝑥𝐴 ¬ ∃𝑦𝐴 𝑥𝑦 ↔ ¬ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)
2 ralnex 3062 . . . . . . . . . 10 (∀𝑦𝐴 ¬ 𝑥𝑦 ↔ ¬ ∃𝑦𝐴 𝑥𝑦)
32rexbii 3084 . . . . . . . . 9 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦 ↔ ∃𝑥𝐴 ¬ ∃𝑦𝐴 𝑥𝑦)
4 ssunib 42713 . . . . . . . . . 10 (𝐴 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)
54notbii 319 . . . . . . . . 9 𝐴 𝐴 ↔ ¬ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)
61, 3, 53bitr4ri 303 . . . . . . . 8 𝐴 𝐴 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
7 simpll 765 . . . . . . . . . . . 12 (((𝐴 ⊆ On ∧ 𝐴 ∈ On) ∧ 𝑥𝐴) → 𝐴 ⊆ On)
87sselda 3972 . . . . . . . . . . 11 ((((𝐴 ⊆ On ∧ 𝐴 ∈ On) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → 𝑦 ∈ On)
9 simpl 481 . . . . . . . . . . . . 13 ((𝐴 ⊆ On ∧ 𝐴 ∈ On) → 𝐴 ⊆ On)
109sselda 3972 . . . . . . . . . . . 12 (((𝐴 ⊆ On ∧ 𝐴 ∈ On) ∧ 𝑥𝐴) → 𝑥 ∈ On)
1110adantr 479 . . . . . . . . . . 11 ((((𝐴 ⊆ On ∧ 𝐴 ∈ On) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → 𝑥 ∈ On)
12 ontri1 6398 . . . . . . . . . . 11 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦𝑥 ↔ ¬ 𝑥𝑦))
138, 11, 12syl2anc 582 . . . . . . . . . 10 ((((𝐴 ⊆ On ∧ 𝐴 ∈ On) ∧ 𝑥𝐴) ∧ 𝑦𝐴) → (𝑦𝑥 ↔ ¬ 𝑥𝑦))
1413ralbidva 3166 . . . . . . . . 9 (((𝐴 ⊆ On ∧ 𝐴 ∈ On) ∧ 𝑥𝐴) → (∀𝑦𝐴 𝑦𝑥 ↔ ∀𝑦𝐴 ¬ 𝑥𝑦))
1514rexbidva 3167 . . . . . . . 8 ((𝐴 ⊆ On ∧ 𝐴 ∈ On) → (∃𝑥𝐴𝑦𝐴 𝑦𝑥 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦))
166, 15bitr4id 289 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴 ∈ On) → (¬ 𝐴 𝐴 ↔ ∃𝑥𝐴𝑦𝐴 𝑦𝑥))
17 unielid 42712 . . . . . . . . 9 ( 𝐴𝐴 ↔ ∃𝑥𝐴𝑦𝐴 𝑦𝑥)
1817a1i 11 . . . . . . . 8 ((𝐴 ⊆ On ∧ 𝐴 ∈ On) → ( 𝐴𝐴 ↔ ∃𝑥𝐴𝑦𝐴 𝑦𝑥))
1918biimprd 247 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴 ∈ On) → (∃𝑥𝐴𝑦𝐴 𝑦𝑥 𝐴𝐴))
2016, 19sylbid 239 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴 ∈ On) → (¬ 𝐴 𝐴 𝐴𝐴))
2120con1d 145 . . . . 5 ((𝐴 ⊆ On ∧ 𝐴 ∈ On) → (¬ 𝐴𝐴𝐴 𝐴))
22 uniss 4911 . . . . 5 (𝐴 𝐴 𝐴 𝐴)
2321, 22syl6 35 . . . 4 ((𝐴 ⊆ On ∧ 𝐴 ∈ On) → (¬ 𝐴𝐴 𝐴 𝐴))
24 ssorduni 7779 . . . . . . . 8 (𝐴 ⊆ On → Ord 𝐴)
25 orduniss 6461 . . . . . . . 8 (Ord 𝐴 𝐴 𝐴)
2624, 25syl 17 . . . . . . 7 (𝐴 ⊆ On → 𝐴 𝐴)
2726biantrud 530 . . . . . 6 (𝐴 ⊆ On → ( 𝐴 𝐴 ↔ ( 𝐴 𝐴 𝐴 𝐴)))
28 eqss 3988 . . . . . 6 ( 𝐴 = 𝐴 ↔ ( 𝐴 𝐴 𝐴 𝐴))
2927, 28bitr4di 288 . . . . 5 (𝐴 ⊆ On → ( 𝐴 𝐴 𝐴 = 𝐴))
3029adantr 479 . . . 4 ((𝐴 ⊆ On ∧ 𝐴 ∈ On) → ( 𝐴 𝐴 𝐴 = 𝐴))
3123, 30sylibd 238 . . 3 ((𝐴 ⊆ On ∧ 𝐴 ∈ On) → (¬ 𝐴𝐴 𝐴 = 𝐴))
3231ex 411 . 2 (𝐴 ⊆ On → ( 𝐴 ∈ On → (¬ 𝐴𝐴 𝐴 = 𝐴)))
33 unon 7832 . . . . 5 On = On
3433a1i 11 . . . 4 ( 𝐴 = On → On = On)
35 unieq 4914 . . . 4 ( 𝐴 = On → 𝐴 = On)
36 id 22 . . . 4 ( 𝐴 = On → 𝐴 = On)
3734, 35, 363eqtr4rd 2776 . . 3 ( 𝐴 = On → 𝐴 = 𝐴)
3837a1i13 27 . 2 (𝐴 ⊆ On → ( 𝐴 = On → (¬ 𝐴𝐴 𝐴 = 𝐴)))
39 ordeleqon 7782 . . 3 (Ord 𝐴 ↔ ( 𝐴 ∈ On ∨ 𝐴 = On))
4024, 39sylib 217 . 2 (𝐴 ⊆ On → ( 𝐴 ∈ On ∨ 𝐴 = On))
4132, 38, 40mpjaod 858 1 (𝐴 ⊆ On → (¬ 𝐴𝐴 𝐴 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845   = wceq 1533  wcel 2098  wral 3051  wrex 3060  wss 3939   cuni 4903  Ord word 6363  Oncon0 6364
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-tr 5261  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6367  df-on 6368  df-suc 6370
This theorem is referenced by:  onsupeqnmax  42740  onsupsucismax  42773
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