nnsseleq - Intuitionistic Logic Explorer

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Theorem nnsseleq 6469

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 6464	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝐴))
2		nntri3or 6461	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
3		df-3or 969	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ∨ 𝐵 ∈ 𝐴))
4	2, 3	sylib 121	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ∨ 𝐵 ∈ 𝐴))
5	4	orcomd 719	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ∈ 𝐴 ∨ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
6	5	ord 714	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (¬ 𝐵 ∈ 𝐴 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
7	1, 6	sylbid 149	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
8		nnord 4589	. . . . 5 ⊢ (𝐵 ∈ ω → Ord 𝐵)
9	8	adantl 275	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → Ord 𝐵)
10		ordelss 4357	. . . . 5 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ⊆ 𝐵)
11	10	ex 114	. . . 4 ⊢ (Ord 𝐵 → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵))
12	9, 11	syl 14	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 → 𝐴 ⊆ 𝐵))
13		eqimss 3196	. . . 4 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
14	13	a1i 9	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵))
15	12, 14	jaod 707	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) → 𝐴 ⊆ 𝐵))
16	7, 15	impbid 128	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 ∨ w3o 967 = wceq 1343 ∈ wcel 2136 ⊆ wss 3116 Ord word 4340 ωcom 4567

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565

This theorem depends on definitions: df-bi 116 df-3or 969 df-3an 970 df-tru 1346 df-nf 1449 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-ral 2449 df-rex 2450 df-v 2728 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-pw 3561 df-sn 3582 df-pr 3583 df-uni 3790 df-int 3825 df-tr 4081 df-iord 4344 df-on 4346 df-suc 4349 df-iom 4568

This theorem is referenced by: nnsssuc 6470 frec2uzled 10364

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