nnsseleq - Intuitionistic Logic Explorer

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Theorem nnsseleq 6520

Description: For natural numbers, inclusion is equivalent to membership or equality. (Contributed by Jim Kingdon, 16-Sep-2021.)

**Assertion**
Ref	Expression
nnsseleq	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))

**Proof of Theorem nnsseleq**
Step	Hyp	Ref	Expression
1		nntri1 6515	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴))
2		nntri3or 6512	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
3		df-3or 981	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ∨ 𝐵 ∈ 𝐴))
4	2, 3	sylib 122	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ∨ 𝐵 ∈ 𝐴))
5	4	orcomd 730	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ∈ 𝐴 ∨ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
6	5	ord 725	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (¬ 𝐵 ∈ 𝐴 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
7	1, 6	sylbid 150	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
8		nnord 4626	. . . . 5 ⊢ (𝐵 ∈ ω → Ord 𝐵)
9	8	adantl 277	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → Ord 𝐵)
10		ordelss 4394	. . . . 5 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ⊆ 𝐵)
11	10	ex 115	. . . 4 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵))
12	9, 11	syl 14	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵))
13		eqimss 3224	. . . 4 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
14	13	a1i 9	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵))
15	12, 14	jaod 718	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) → 𝐴 ⊆ 𝐵))
16	7, 15	impbid 129	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 ∨ w3o 979 = wceq 1364 ∈ wcel 2160 ⊆ wss 3144 Ord word 4377 ωcom 4604

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-uni 3825 df-int 3860 df-tr 4117 df-iord 4381 df-on 4383 df-suc 4386 df-iom 4605

This theorem is referenced by: nnsssuc 6521 frec2uzled 10448

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