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Theorem naddssim 33480
Description: Ordinal less-than-or-equal is preserved by natural addition. (Contributed by Scott Fenton, 7-Sep-2024.)
Assertion
Ref Expression
naddssim ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶)))

Proof of Theorem naddssim
Dummy variables 𝑐 𝑑 𝑤 𝑥 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7178 . . . . . . 7 (𝑐 = 𝑑 → (𝐴 +no 𝑐) = (𝐴 +no 𝑑))
2 oveq2 7178 . . . . . . 7 (𝑐 = 𝑑 → (𝐵 +no 𝑐) = (𝐵 +no 𝑑))
31, 2sseq12d 3910 . . . . . 6 (𝑐 = 𝑑 → ((𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐) ↔ (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)))
43imbi2d 344 . . . . 5 (𝑐 = 𝑑 → ((𝐴𝐵 → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐)) ↔ (𝐴𝐵 → (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑))))
54imbi2d 344 . . . 4 (𝑐 = 𝑑 → (((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐))) ↔ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)))))
6 oveq2 7178 . . . . . . 7 (𝑐 = 𝐶 → (𝐴 +no 𝑐) = (𝐴 +no 𝐶))
7 oveq2 7178 . . . . . . 7 (𝑐 = 𝐶 → (𝐵 +no 𝑐) = (𝐵 +no 𝐶))
86, 7sseq12d 3910 . . . . . 6 (𝑐 = 𝐶 → ((𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐) ↔ (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶)))
98imbi2d 344 . . . . 5 (𝑐 = 𝐶 → ((𝐴𝐵 → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐)) ↔ (𝐴𝐵 → (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶))))
109imbi2d 344 . . . 4 (𝑐 = 𝐶 → (((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐))) ↔ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶)))))
11 r19.21v 3089 . . . . . 6 (∀𝑑𝑐 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑))) ↔ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∀𝑑𝑐 (𝐴𝐵 → (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑))))
12 r19.21v 3089 . . . . . . 7 (∀𝑑𝑐 (𝐴𝐵 → (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ↔ (𝐴𝐵 → ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)))
1312imbi2i 339 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∀𝑑𝑐 (𝐴𝐵 → (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑))) ↔ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑))))
1411, 13bitri 278 . . . . 5 (∀𝑑𝑐 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑))) ↔ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑))))
15 oveq2 7178 . . . . . . . . . . . . . . . . . . 19 (𝑑 = 𝑤 → (𝐴 +no 𝑑) = (𝐴 +no 𝑤))
16 oveq2 7178 . . . . . . . . . . . . . . . . . . 19 (𝑑 = 𝑤 → (𝐵 +no 𝑑) = (𝐵 +no 𝑤))
1715, 16sseq12d 3910 . . . . . . . . . . . . . . . . . 18 (𝑑 = 𝑤 → ((𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑) ↔ (𝐴 +no 𝑤) ⊆ (𝐵 +no 𝑤)))
1817rspccva 3525 . . . . . . . . . . . . . . . . 17 ((∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑) ∧ 𝑤𝑐) → (𝐴 +no 𝑤) ⊆ (𝐵 +no 𝑤))
1918ad4ant24 754 . . . . . . . . . . . . . . . 16 ((((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) ∧ 𝑤𝑐) → (𝐴 +no 𝑤) ⊆ (𝐵 +no 𝑤))
20 simprrl 781 . . . . . . . . . . . . . . . . 17 (((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) → ∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥)
21 oveq2 7178 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑤 → (𝐵 +no 𝑦) = (𝐵 +no 𝑤))
2221eleq1d 2817 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑤 → ((𝐵 +no 𝑦) ∈ 𝑥 ↔ (𝐵 +no 𝑤) ∈ 𝑥))
2322rspccva 3525 . . . . . . . . . . . . . . . . 17 ((∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥𝑤𝑐) → (𝐵 +no 𝑤) ∈ 𝑥)
2420, 23sylan 583 . . . . . . . . . . . . . . . 16 ((((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) ∧ 𝑤𝑐) → (𝐵 +no 𝑤) ∈ 𝑥)
25 simplrl 777 . . . . . . . . . . . . . . . . . . . . 21 (((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) → 𝐴 ∈ On)
2625adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → 𝐴 ∈ On)
2726adantr 484 . . . . . . . . . . . . . . . . . . 19 (((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) → 𝐴 ∈ On)
2827adantr 484 . . . . . . . . . . . . . . . . . 18 ((((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) ∧ 𝑤𝑐) → 𝐴 ∈ On)
29 simp-4l 783 . . . . . . . . . . . . . . . . . . 19 (((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) → 𝑐 ∈ On)
30 onelon 6197 . . . . . . . . . . . . . . . . . . 19 ((𝑐 ∈ On ∧ 𝑤𝑐) → 𝑤 ∈ On)
3129, 30sylan 583 . . . . . . . . . . . . . . . . . 18 ((((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) ∧ 𝑤𝑐) → 𝑤 ∈ On)
32 naddcl 33475 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ On ∧ 𝑤 ∈ On) → (𝐴 +no 𝑤) ∈ On)
3328, 31, 32syl2anc 587 . . . . . . . . . . . . . . . . 17 ((((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) ∧ 𝑤𝑐) → (𝐴 +no 𝑤) ∈ On)
34 simplrl 777 . . . . . . . . . . . . . . . . 17 ((((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) ∧ 𝑤𝑐) → 𝑥 ∈ On)
35 ontr2 6219 . . . . . . . . . . . . . . . . 17 (((𝐴 +no 𝑤) ∈ On ∧ 𝑥 ∈ On) → (((𝐴 +no 𝑤) ⊆ (𝐵 +no 𝑤) ∧ (𝐵 +no 𝑤) ∈ 𝑥) → (𝐴 +no 𝑤) ∈ 𝑥))
3633, 34, 35syl2anc 587 . . . . . . . . . . . . . . . 16 ((((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) ∧ 𝑤𝑐) → (((𝐴 +no 𝑤) ⊆ (𝐵 +no 𝑤) ∧ (𝐵 +no 𝑤) ∈ 𝑥) → (𝐴 +no 𝑤) ∈ 𝑥))
3719, 24, 36mp2and 699 . . . . . . . . . . . . . . 15 ((((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) ∧ 𝑤𝑐) → (𝐴 +no 𝑤) ∈ 𝑥)
3837ralrimiva 3096 . . . . . . . . . . . . . 14 (((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) → ∀𝑤𝑐 (𝐴 +no 𝑤) ∈ 𝑥)
39 simpllr 776 . . . . . . . . . . . . . . 15 (((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) → 𝐴𝐵)
40 simprrr 782 . . . . . . . . . . . . . . 15 (((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) → ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥)
41 ssralv 3943 . . . . . . . . . . . . . . 15 (𝐴𝐵 → (∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥 → ∀𝑧𝐴 (𝑧 +no 𝑐) ∈ 𝑥))
4239, 40, 41sylc 65 . . . . . . . . . . . . . 14 (((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) → ∀𝑧𝐴 (𝑧 +no 𝑐) ∈ 𝑥)
4338, 42jca 515 . . . . . . . . . . . . 13 (((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) → (∀𝑤𝑐 (𝐴 +no 𝑤) ∈ 𝑥 ∧ ∀𝑧𝐴 (𝑧 +no 𝑐) ∈ 𝑥))
4443expr 460 . . . . . . . . . . . 12 (((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ 𝑥 ∈ On) → ((∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥) → (∀𝑤𝑐 (𝐴 +no 𝑤) ∈ 𝑥 ∧ ∀𝑧𝐴 (𝑧 +no 𝑐) ∈ 𝑥)))
4544ss2rabdv 3965 . . . . . . . . . . 11 ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → {𝑥 ∈ On ∣ (∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥)} ⊆ {𝑥 ∈ On ∣ (∀𝑤𝑐 (𝐴 +no 𝑤) ∈ 𝑥 ∧ ∀𝑧𝐴 (𝑧 +no 𝑐) ∈ 𝑥)})
46 intss 4857 . . . . . . . . . . 11 ({𝑥 ∈ On ∣ (∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥)} ⊆ {𝑥 ∈ On ∣ (∀𝑤𝑐 (𝐴 +no 𝑤) ∈ 𝑥 ∧ ∀𝑧𝐴 (𝑧 +no 𝑐) ∈ 𝑥)} → {𝑥 ∈ On ∣ (∀𝑤𝑐 (𝐴 +no 𝑤) ∈ 𝑥 ∧ ∀𝑧𝐴 (𝑧 +no 𝑐) ∈ 𝑥)} ⊆ {𝑥 ∈ On ∣ (∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥)})
4745, 46syl 17 . . . . . . . . . 10 ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → {𝑥 ∈ On ∣ (∀𝑤𝑐 (𝐴 +no 𝑤) ∈ 𝑥 ∧ ∀𝑧𝐴 (𝑧 +no 𝑐) ∈ 𝑥)} ⊆ {𝑥 ∈ On ∣ (∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥)})
48 simplll 775 . . . . . . . . . . 11 ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → 𝑐 ∈ On)
49 naddov2 33477 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑐 ∈ On) → (𝐴 +no 𝑐) = {𝑥 ∈ On ∣ (∀𝑤𝑐 (𝐴 +no 𝑤) ∈ 𝑥 ∧ ∀𝑧𝐴 (𝑧 +no 𝑐) ∈ 𝑥)})
5026, 48, 49syl2anc 587 . . . . . . . . . 10 ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → (𝐴 +no 𝑐) = {𝑥 ∈ On ∣ (∀𝑤𝑐 (𝐴 +no 𝑤) ∈ 𝑥 ∧ ∀𝑧𝐴 (𝑧 +no 𝑐) ∈ 𝑥)})
51 simplrr 778 . . . . . . . . . . . 12 (((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) → 𝐵 ∈ On)
5251adantr 484 . . . . . . . . . . 11 ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → 𝐵 ∈ On)
53 naddov2 33477 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑐 ∈ On) → (𝐵 +no 𝑐) = {𝑥 ∈ On ∣ (∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥)})
5452, 48, 53syl2anc 587 . . . . . . . . . 10 ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → (𝐵 +no 𝑐) = {𝑥 ∈ On ∣ (∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥)})
5547, 50, 543sstr4d 3924 . . . . . . . . 9 ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐))
5655exp31 423 . . . . . . . 8 ((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴𝐵 → (∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑) → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐))))
5756a2d 29 . . . . . . 7 ((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴𝐵 → ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → (𝐴𝐵 → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐))))
5857ex 416 . . . . . 6 (𝑐 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴𝐵 → ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → (𝐴𝐵 → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐)))))
5958a2d 29 . . . . 5 (𝑐 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑))) → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐)))))
6014, 59syl5bi 245 . . . 4 (𝑐 ∈ On → (∀𝑑𝑐 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑))) → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐)))))
615, 10, 60tfis3 7591 . . 3 (𝐶 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶))))
6261com12 32 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ On → (𝐴𝐵 → (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶))))
63623impia 1118 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3053  {crab 3057  wss 3843   cint 4836  Oncon0 6172  (class class class)co 7170   +no cnadd 33467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-1st 7714  df-2nd 7715  df-frecs 33438  df-nadd 33468
This theorem is referenced by:  naddel1  33482
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