ordeldifsucon - Mathbox for Richard Penner

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Theorem ordeldifsucon 43800

Description: Membership in the difference of ordinal and successor ordinal. (Contributed by RP, 16-Jan-2025.)

**Assertion**
Ref	Expression
ordeldifsucon	⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐶 ∈ (𝐴 ∖ suc 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶)))

**Proof of Theorem ordeldifsucon**
Step	Hyp	Ref	Expression
1		eldif 3914	. 2 ⊢ (𝐶 ∈ (𝐴 ∖ suc 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ∈ suc 𝐵))
2		simplr 778	. . . . 5 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ On)
3		ordelord 6364	. . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝐶 ∈ 𝐴) → Ord 𝐶)
4	3	adantlr 725	. . . . 5 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ 𝐴) → Ord 𝐶)
5		ordelsuc 7796	. . . . 5 ⊢ ((𝐵 ∈ On ∧ Ord 𝐶) → (𝐵 ∈ 𝐶 ↔ suc 𝐵 ⊆ 𝐶))
6	2, 4, 5	syl2anc 593	. . . 4 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ 𝐴) → (𝐵 ∈ 𝐶 ↔ suc 𝐵 ⊆ 𝐶))
7		eloni 6352	. . . . . 6 ⊢ (𝐵 ∈ On → Ord 𝐵)
8		ordsuci 7787	. . . . . 6 ⊢ (Ord 𝐵 → Ord suc 𝐵)
9	2, 7, 8	3syl 18	. . . . 5 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ 𝐴) → Ord suc 𝐵)
10		ordtri1 6375	. . . . 5 ⊢ ((Ord suc 𝐵 ∧ Ord 𝐶) → (suc 𝐵 ⊆ 𝐶 ↔ ¬ 𝐶 ∈ suc 𝐵))
11	9, 4, 10	syl2anc 593	. . . 4 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ 𝐴) → (suc 𝐵 ⊆ 𝐶 ↔ ¬ 𝐶 ∈ suc 𝐵))
12	6, 11	bitr2d 282	. . 3 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ 𝐴) → (¬ 𝐶 ∈ suc 𝐵 ↔ 𝐵 ∈ 𝐶))
13	12	pm5.32da 587	. 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On) → ((𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ∈ suc 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶)))
14	1, 13	bitrid 285	1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐶 ∈ (𝐴 ∖ suc 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∈ wcel 2141 ∖ cdif 3901 ⊆ wss 3904 Ord word 6341 Oncon0 6342 suc csuc 6344

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-ext 2733 ax-sep 5245 ax-pr 5389

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-sb 2090 df-clab 2740 df-cleq 2753 df-clel 2836 df-ne 2957 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-br 5100 df-opab 5162 df-tr 5207 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-ord 6345 df-on 6346 df-suc 6348

This theorem is referenced by: orddif0suc 43809 cantnfresb 43865