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Theorem onmindif 5977
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 3726 . . . 4 (𝑥 ∈ (𝐴 ∖ suc 𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ suc 𝐵))
2 ssel2 3740 . . . . . . . . 9 ((𝐴 ⊆ On ∧ 𝑥𝐴) → 𝑥 ∈ On)
3 ontri1 5919 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝐵 ↔ ¬ 𝐵𝑥))
4 onsssuc 5975 . . . . . . . . . . 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 489 . . . . . . . 8 (((𝐴 ⊆ On ∧ 𝑥𝐴) ∧ 𝐵 ∈ On) → (¬ 𝑥 ∈ suc 𝐵𝐵𝑥))
87biimpd 219 . . . . . . 7 (((𝐴 ⊆ On ∧ 𝑥𝐴) ∧ 𝐵 ∈ On) → (¬ 𝑥 ∈ suc 𝐵𝐵𝑥))
98exp31 631 . . . . . 6 (𝐴 ⊆ On → (𝑥𝐴 → (𝐵 ∈ On → (¬ 𝑥 ∈ suc 𝐵𝐵𝑥))))
109com23 86 . . . . 5 (𝐴 ⊆ On → (𝐵 ∈ On → (𝑥𝐴 → (¬ 𝑥 ∈ suc 𝐵𝐵𝑥))))
1110imp4b 614 . . . 4 ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → ((𝑥𝐴 ∧ ¬ 𝑥 ∈ suc 𝐵) → 𝐵𝑥))
121, 11syl5bi 232 . . 3 ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → (𝑥 ∈ (𝐴 ∖ suc 𝐵) → 𝐵𝑥))
1312ralrimiv 3104 . 2 ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → ∀𝑥 ∈ (𝐴 ∖ suc 𝐵)𝐵𝑥)
14 elintg 4636 . . 3 (𝐵 ∈ On → (𝐵 (𝐴 ∖ suc 𝐵) ↔ ∀𝑥 ∈ (𝐴 ∖ suc 𝐵)𝐵𝑥))
1514adantl 473 . 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 2140  wral 3051  cdif 3713  wss 3716   cint 4628  Oncon0 5885  suc csuc 5887
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 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
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 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-int 4629  df-br 4806  df-opab 4866  df-tr 4906  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-ord 5888  df-on 5889  df-suc 5891
This theorem is referenced by:  unblem3  8382  fin23lem26  9360
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