ordeldifsucon - Mathbox for Richard Penner

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Theorem ordeldifsucon 43501

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 3911	. 2 ⊢ (𝐶 ∈ (𝐴 ∖ suc 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ∈ suc 𝐵))
2		simplr 768	. . . . 5 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ On)
3		ordelord 6339	. . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝐶 ∈ 𝐴) → Ord 𝐶)
4	3	adantlr 715	. . . . 5 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ 𝐴) → Ord 𝐶)
5		ordelsuc 7762	. . . . 5 ⊢ ((𝐵 ∈ On ∧ Ord 𝐶) → (𝐵 ∈ 𝐶 ↔ suc 𝐵 ⊆ 𝐶))
6	2, 4, 5	syl2anc 584	. . . 4 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ 𝐴) → (𝐵 ∈ 𝐶 ↔ suc 𝐵 ⊆ 𝐶))
7		eloni 6327	. . . . . 6 ⊢ (𝐵 ∈ On → Ord 𝐵)
8		ordsuci 7753	. . . . . 6 ⊢ (Ord 𝐵 → Ord suc 𝐵)
9	2, 7, 8	3syl 18	. . . . 5 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ 𝐴) → Ord suc 𝐵)
10		ordtri1 6350	. . . . 5 ⊢ ((Ord suc 𝐵 ∧ Ord 𝐶) → (suc 𝐵 ⊆ 𝐶 ↔ ¬ 𝐶 ∈ suc 𝐵))
11	9, 4, 10	syl2anc 584	. . . 4 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ 𝐴) → (suc 𝐵 ⊆ 𝐶 ↔ ¬ 𝐶 ∈ suc 𝐵))
12	6, 11	bitr2d 280	. . 3 ⊢ (((Ord 𝐴 ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ 𝐴) → (¬ 𝐶 ∈ suc 𝐵 ↔ 𝐵 ∈ 𝐶))
13	12	pm5.32da 579	. 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On) → ((𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ∈ suc 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶)))
14	1, 13	bitrid 283	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 206 ∧ wa 395 ∈ wcel 2113 ∖ cdif 3898 ⊆ wss 3901 Ord word 6316 Oncon0 6317 suc csuc 6319

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pr 5377

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-sb 2068 df-clab 2715 df-cleq 2728 df-clel 2811 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-tr 5206 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-ord 6320 df-on 6321 df-suc 6323

This theorem is referenced by: orddif0suc 43510 cantnfresb 43566