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Theorem onmindif 5853
Description: When its successor is subtracted from a class of ordinal numbers, an ordinal number is less than the minimum of the resulting subclass. (Contributed by NM, 1-Dec-2003.)
Assertion
Ref Expression
onmindif ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → 𝐵 (𝐴 ∖ suc 𝐵))

Proof of Theorem onmindif
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eldif 3617 . . . 4 (𝑥 ∈ (𝐴 ∖ suc 𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ suc 𝐵))
2 ssel2 3631 . . . . . . . . 9 ((𝐴 ⊆ On ∧ 𝑥𝐴) → 𝑥 ∈ On)
3 ontri1 5795 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝐵 ↔ ¬ 𝐵𝑥))
4 onsssuc 5851 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝐵𝑥 ∈ suc 𝐵))
53, 4bitr3d 270 . . . . . . . . . 10 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐵𝑥𝑥 ∈ suc 𝐵))
65con1bid 344 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝑥 ∈ suc 𝐵𝐵𝑥))
72, 6sylan 487 . . . . . . . 8 (((𝐴 ⊆ On ∧ 𝑥𝐴) ∧ 𝐵 ∈ On) → (¬ 𝑥 ∈ suc 𝐵𝐵𝑥))
87biimpd 219 . . . . . . 7 (((𝐴 ⊆ On ∧ 𝑥𝐴) ∧ 𝐵 ∈ On) → (¬ 𝑥 ∈ suc 𝐵𝐵𝑥))
98exp31 629 . . . . . 6 (𝐴 ⊆ On → (𝑥𝐴 → (𝐵 ∈ On → (¬ 𝑥 ∈ suc 𝐵𝐵𝑥))))
109com23 86 . . . . 5 (𝐴 ⊆ On → (𝐵 ∈ On → (𝑥𝐴 → (¬ 𝑥 ∈ suc 𝐵𝐵𝑥))))
1110imp4b 612 . . . 4 ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → ((𝑥𝐴 ∧ ¬ 𝑥 ∈ suc 𝐵) → 𝐵𝑥))
121, 11syl5bi 232 . . 3 ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → (𝑥 ∈ (𝐴 ∖ suc 𝐵) → 𝐵𝑥))
1312ralrimiv 2994 . 2 ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → ∀𝑥 ∈ (𝐴 ∖ suc 𝐵)𝐵𝑥)
14 elintg 4515 . . 3 (𝐵 ∈ On → (𝐵 (𝐴 ∖ suc 𝐵) ↔ ∀𝑥 ∈ (𝐴 ∖ suc 𝐵)𝐵𝑥))
1514adantl 481 . 2 ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → (𝐵 (𝐴 ∖ suc 𝐵) ↔ ∀𝑥 ∈ (𝐴 ∖ suc 𝐵)𝐵𝑥))
1613, 15mpbird 247 1 ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → 𝐵 (𝐴 ∖ suc 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wcel 2030  wral 2941  cdif 3604  wss 3607   cint 4507  Oncon0 5761  suc csuc 5763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-suc 5767
This theorem is referenced by:  unblem3  8255  fin23lem26  9185
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