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Theorem ordunisuc2 7210
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 7209 . 2 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
2 ralnex 3130 . . 3 (∀𝑥 ∈ On ¬ 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
3 suceloni 7179 . . . . . . . . . 10 (𝑥 ∈ On → suc 𝑥 ∈ On)
4 eloni 5894 . . . . . . . . . 10 (suc 𝑥 ∈ On → Ord suc 𝑥)
53, 4syl 17 . . . . . . . . 9 (𝑥 ∈ On → Ord suc 𝑥)
6 ordtri3 5920 . . . . . . . . 9 ((Ord 𝐴 ∧ Ord suc 𝑥) → (𝐴 = suc 𝑥 ↔ ¬ (𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴)))
75, 6sylan2 492 . . . . . . . 8 ((Ord 𝐴𝑥 ∈ On) → (𝐴 = suc 𝑥 ↔ ¬ (𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴)))
87con2bid 343 . . . . . . 7 ((Ord 𝐴𝑥 ∈ On) → ((𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴) ↔ ¬ 𝐴 = suc 𝑥))
9 onnbtwn 5979 . . . . . . . . . . . . 13 (𝑥 ∈ On → ¬ (𝑥𝐴𝐴 ∈ suc 𝑥))
10 imnan 437 . . . . . . . . . . . . 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 473 . . . . . . . . 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 394 . . . . . . . 8 ((Ord 𝐴𝑥 ∈ On) → ((𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴) → (𝑥𝐴 → suc 𝑥𝐴)))
19 eloni 5894 . . . . . . . . . . . . . 14 (𝑥 ∈ On → Ord 𝑥)
20 ordtri2or 5983 . . . . . . . . . . . . . 14 ((Ord 𝑥 ∧ Ord 𝐴) → (𝑥𝐴𝐴𝑥))
2119, 20sylan 489 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ Ord 𝐴) → (𝑥𝐴𝐴𝑥))
2221ancoms 468 . . . . . . . . . . . 12 ((Ord 𝐴𝑥 ∈ On) → (𝑥𝐴𝐴𝑥))
2322orcomd 402 . . . . . . . . . . 11 ((Ord 𝐴𝑥 ∈ On) → (𝐴𝑥𝑥𝐴))
2423adantr 472 . . . . . . . . . 10 (((Ord 𝐴𝑥 ∈ On) ∧ (𝑥𝐴 → suc 𝑥𝐴)) → (𝐴𝑥𝑥𝐴))
25 ordsssuc2 5975 . . . . . . . . . . . . 13 ((Ord 𝐴𝑥 ∈ On) → (𝐴𝑥𝐴 ∈ suc 𝑥))
2625biimpd 219 . . . . . . . . . . . 12 ((Ord 𝐴𝑥 ∈ On) → (𝐴𝑥𝐴 ∈ suc 𝑥))
2726adantr 472 . . . . . . . . . . 11 (((Ord 𝐴𝑥 ∈ On) ∧ (𝑥𝐴 → suc 𝑥𝐴)) → (𝐴𝑥𝐴 ∈ suc 𝑥))
28 simpr 479 . . . . . . . . . . 11 (((Ord 𝐴𝑥 ∈ On) ∧ (𝑥𝐴 → suc 𝑥𝐴)) → (𝑥𝐴 → suc 𝑥𝐴))
2927, 28orim12d 919 . . . . . . . . . 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 449 . . . . . . . 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 725 . . . . 5 (Ord 𝐴 → ((𝑥 ∈ On → ¬ 𝐴 = suc 𝑥) ↔ (𝑥 ∈ On → (𝑥𝐴 → suc 𝑥𝐴))))
35 impexp 461 . . . . . 6 (((𝑥 ∈ On ∧ 𝑥𝐴) → suc 𝑥𝐴) ↔ (𝑥 ∈ On → (𝑥𝐴 → suc 𝑥𝐴)))
36 simpr 479 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑥𝐴) → 𝑥𝐴)
37 ordelon 5908 . . . . . . . . . 10 ((Ord 𝐴𝑥𝐴) → 𝑥 ∈ On)
3837ex 449 . . . . . . . . 9 (Ord 𝐴 → (𝑥𝐴𝑥 ∈ On))
3938ancrd 578 . . . . . . . 8 (Ord 𝐴 → (𝑥𝐴 → (𝑥 ∈ On ∧ 𝑥𝐴)))
4036, 39impbid2 216 . . . . . . 7 (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥𝐴) ↔ 𝑥𝐴))
4140imbi1d 330 . . . . . 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 3122 . . 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 382  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  wss 3715   cuni 4588  Ord word 5883  Oncon0 5884  suc csuc 5886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-tr 4905  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-ord 5887  df-on 5888  df-suc 5890
This theorem is referenced by:  dflim4  7214  limsuc2  38131
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