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Theorem ordunisuc2 7006
Description: An ordinal equal to its union contains the successor of each of its members. (Contributed by NM, 1-Feb-2005.)
Assertion
Ref Expression
ordunisuc2 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ∀𝑥𝐴 suc 𝑥𝐴))
Distinct variable group:   𝑥,𝐴

Proof of Theorem ordunisuc2
StepHypRef Expression
1 orduninsuc 7005 . 2 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
2 ralnex 2988 . . 3 (∀𝑥 ∈ On ¬ 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
3 suceloni 6975 . . . . . . . . . 10 (𝑥 ∈ On → suc 𝑥 ∈ On)
4 eloni 5702 . . . . . . . . . 10 (suc 𝑥 ∈ On → Ord suc 𝑥)
53, 4syl 17 . . . . . . . . 9 (𝑥 ∈ On → Ord suc 𝑥)
6 ordtri3 5728 . . . . . . . . 9 ((Ord 𝐴 ∧ Ord suc 𝑥) → (𝐴 = suc 𝑥 ↔ ¬ (𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴)))
75, 6sylan2 491 . . . . . . . 8 ((Ord 𝐴𝑥 ∈ On) → (𝐴 = suc 𝑥 ↔ ¬ (𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴)))
87con2bid 344 . . . . . . 7 ((Ord 𝐴𝑥 ∈ On) → ((𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴) ↔ ¬ 𝐴 = suc 𝑥))
9 onnbtwn 5787 . . . . . . . . . . . . 13 (𝑥 ∈ On → ¬ (𝑥𝐴𝐴 ∈ suc 𝑥))
10 imnan 438 . . . . . . . . . . . . 13 ((𝑥𝐴 → ¬ 𝐴 ∈ suc 𝑥) ↔ ¬ (𝑥𝐴𝐴 ∈ suc 𝑥))
119, 10sylibr 224 . . . . . . . . . . . 12 (𝑥 ∈ On → (𝑥𝐴 → ¬ 𝐴 ∈ suc 𝑥))
1211con2d 129 . . . . . . . . . . 11 (𝑥 ∈ On → (𝐴 ∈ suc 𝑥 → ¬ 𝑥𝐴))
13 pm2.21 120 . . . . . . . . . . 11 𝑥𝐴 → (𝑥𝐴 → suc 𝑥𝐴))
1412, 13syl6 35 . . . . . . . . . 10 (𝑥 ∈ On → (𝐴 ∈ suc 𝑥 → (𝑥𝐴 → suc 𝑥𝐴)))
1514adantl 482 . . . . . . . . 9 ((Ord 𝐴𝑥 ∈ On) → (𝐴 ∈ suc 𝑥 → (𝑥𝐴 → suc 𝑥𝐴)))
16 ax-1 6 . . . . . . . . . 10 (suc 𝑥𝐴 → (𝑥𝐴 → suc 𝑥𝐴))
1716a1i 11 . . . . . . . . 9 ((Ord 𝐴𝑥 ∈ On) → (suc 𝑥𝐴 → (𝑥𝐴 → suc 𝑥𝐴)))
1815, 17jaod 395 . . . . . . . 8 ((Ord 𝐴𝑥 ∈ On) → ((𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴) → (𝑥𝐴 → suc 𝑥𝐴)))
19 eloni 5702 . . . . . . . . . . . . . 14 (𝑥 ∈ On → Ord 𝑥)
20 ordtri2or 5791 . . . . . . . . . . . . . 14 ((Ord 𝑥 ∧ Ord 𝐴) → (𝑥𝐴𝐴𝑥))
2119, 20sylan 488 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ Ord 𝐴) → (𝑥𝐴𝐴𝑥))
2221ancoms 469 . . . . . . . . . . . 12 ((Ord 𝐴𝑥 ∈ On) → (𝑥𝐴𝐴𝑥))
2322orcomd 403 . . . . . . . . . . 11 ((Ord 𝐴𝑥 ∈ On) → (𝐴𝑥𝑥𝐴))
2423adantr 481 . . . . . . . . . 10 (((Ord 𝐴𝑥 ∈ On) ∧ (𝑥𝐴 → suc 𝑥𝐴)) → (𝐴𝑥𝑥𝐴))
25 ordsssuc2 5783 . . . . . . . . . . . . 13 ((Ord 𝐴𝑥 ∈ On) → (𝐴𝑥𝐴 ∈ suc 𝑥))
2625biimpd 219 . . . . . . . . . . . 12 ((Ord 𝐴𝑥 ∈ On) → (𝐴𝑥𝐴 ∈ suc 𝑥))
2726adantr 481 . . . . . . . . . . 11 (((Ord 𝐴𝑥 ∈ On) ∧ (𝑥𝐴 → suc 𝑥𝐴)) → (𝐴𝑥𝐴 ∈ suc 𝑥))
28 simpr 477 . . . . . . . . . . 11 (((Ord 𝐴𝑥 ∈ On) ∧ (𝑥𝐴 → suc 𝑥𝐴)) → (𝑥𝐴 → suc 𝑥𝐴))
2927, 28orim12d 882 . . . . . . . . . 10 (((Ord 𝐴𝑥 ∈ On) ∧ (𝑥𝐴 → suc 𝑥𝐴)) → ((𝐴𝑥𝑥𝐴) → (𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴)))
3024, 29mpd 15 . . . . . . . . 9 (((Ord 𝐴𝑥 ∈ On) ∧ (𝑥𝐴 → suc 𝑥𝐴)) → (𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴))
3130ex 450 . . . . . . . 8 ((Ord 𝐴𝑥 ∈ On) → ((𝑥𝐴 → suc 𝑥𝐴) → (𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴)))
3218, 31impbid 202 . . . . . . 7 ((Ord 𝐴𝑥 ∈ On) → ((𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴) ↔ (𝑥𝐴 → suc 𝑥𝐴)))
338, 32bitr3d 270 . . . . . 6 ((Ord 𝐴𝑥 ∈ On) → (¬ 𝐴 = suc 𝑥 ↔ (𝑥𝐴 → suc 𝑥𝐴)))
3433pm5.74da 722 . . . . 5 (Ord 𝐴 → ((𝑥 ∈ On → ¬ 𝐴 = suc 𝑥) ↔ (𝑥 ∈ On → (𝑥𝐴 → suc 𝑥𝐴))))
35 impexp 462 . . . . . 6 (((𝑥 ∈ On ∧ 𝑥𝐴) → suc 𝑥𝐴) ↔ (𝑥 ∈ On → (𝑥𝐴 → suc 𝑥𝐴)))
36 simpr 477 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑥𝐴) → 𝑥𝐴)
37 ordelon 5716 . . . . . . . . . 10 ((Ord 𝐴𝑥𝐴) → 𝑥 ∈ On)
3837ex 450 . . . . . . . . 9 (Ord 𝐴 → (𝑥𝐴𝑥 ∈ On))
3938ancrd 576 . . . . . . . 8 (Ord 𝐴 → (𝑥𝐴 → (𝑥 ∈ On ∧ 𝑥𝐴)))
4036, 39impbid2 216 . . . . . . 7 (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥𝐴) ↔ 𝑥𝐴))
4140imbi1d 331 . . . . . 6 (Ord 𝐴 → (((𝑥 ∈ On ∧ 𝑥𝐴) → suc 𝑥𝐴) ↔ (𝑥𝐴 → suc 𝑥𝐴)))
4235, 41syl5bbr 274 . . . . 5 (Ord 𝐴 → ((𝑥 ∈ On → (𝑥𝐴 → suc 𝑥𝐴)) ↔ (𝑥𝐴 → suc 𝑥𝐴)))
4334, 42bitrd 268 . . . 4 (Ord 𝐴 → ((𝑥 ∈ On → ¬ 𝐴 = suc 𝑥) ↔ (𝑥𝐴 → suc 𝑥𝐴)))
4443ralbidv2 2980 . . 3 (Ord 𝐴 → (∀𝑥 ∈ On ¬ 𝐴 = suc 𝑥 ↔ ∀𝑥𝐴 suc 𝑥𝐴))
452, 44syl5bbr 274 . 2 (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ∀𝑥𝐴 suc 𝑥𝐴))
461, 45bitrd 268 1 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ∀𝑥𝐴 suc 𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  wral 2908  wrex 2909  wss 3560   cuni 4409  Ord word 5691  Oncon0 5692  suc csuc 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-tr 4723  df-eprel 4995  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-ord 5695  df-on 5696  df-suc 5698
This theorem is referenced by:  dflim4  7010  limsuc2  37130
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