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Theorem naddssim 8630
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 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7365 . . . . . . 7 (𝑐 = 𝑑 → (𝐴 +no 𝑐) = (𝐴 +no 𝑑))
2 oveq2 7365 . . . . . . 7 (𝑐 = 𝑑 → (𝐵 +no 𝑐) = (𝐵 +no 𝑑))
31, 2sseq12d 3977 . . . . . 6 (𝑐 = 𝑑 → ((𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐) ↔ (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)))
43imbi2d 340 . . . . 5 (𝑐 = 𝑑 → ((𝐴𝐵 → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐)) ↔ (𝐴𝐵 → (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑))))
54imbi2d 340 . . . 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 7365 . . . . . . 7 (𝑐 = 𝐶 → (𝐴 +no 𝑐) = (𝐴 +no 𝐶))
7 oveq2 7365 . . . . . . 7 (𝑐 = 𝐶 → (𝐵 +no 𝑐) = (𝐵 +no 𝐶))
86, 7sseq12d 3977 . . . . . 6 (𝑐 = 𝐶 → ((𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐) ↔ (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶)))
98imbi2d 340 . . . . 5 (𝑐 = 𝐶 → ((𝐴𝐵 → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐)) ↔ (𝐴𝐵 → (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶))))
109imbi2d 340 . . . 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 3176 . . . . . 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 3176 . . . . . . 7 (∀𝑑𝑐 (𝐴𝐵 → (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ↔ (𝐴𝐵 → ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)))
1312imbi2i 335 . . . . . 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 274 . . . . 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 7365 . . . . . . . . . . . . . . . . . . 19 (𝑑 = 𝑤 → (𝐴 +no 𝑑) = (𝐴 +no 𝑤))
16 oveq2 7365 . . . . . . . . . . . . . . . . . . 19 (𝑑 = 𝑤 → (𝐵 +no 𝑑) = (𝐵 +no 𝑤))
1715, 16sseq12d 3977 . . . . . . . . . . . . . . . . . 18 (𝑑 = 𝑤 → ((𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑) ↔ (𝐴 +no 𝑤) ⊆ (𝐵 +no 𝑤)))
1817rspccva 3580 . . . . . . . . . . . . . . . . 17 ((∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑) ∧ 𝑤𝑐) → (𝐴 +no 𝑤) ⊆ (𝐵 +no 𝑤))
1918ad4ant24 752 . . . . . . . . . . . . . . . 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 779 . . . . . . . . . . . . . . . . 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 7365 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑤 → (𝐵 +no 𝑦) = (𝐵 +no 𝑤))
2221eleq1d 2822 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑤 → ((𝐵 +no 𝑦) ∈ 𝑥 ↔ (𝐵 +no 𝑤) ∈ 𝑥))
2322rspccva 3580 . . . . . . . . . . . . . . . . 17 ((∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥𝑤𝑐) → (𝐵 +no 𝑤) ∈ 𝑥)
2420, 23sylan 580 . . . . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . . . . . . . . 21 (((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) → 𝐴 ∈ On)
2625adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → 𝐴 ∈ On)
2726adantr 481 . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . 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 781 . . . . . . . . . . . . . . . . . . 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 6342 . . . . . . . . . . . . . . . . . . 19 ((𝑐 ∈ On ∧ 𝑤𝑐) → 𝑤 ∈ On)
3129, 30sylan 580 . . . . . . . . . . . . . . . . . 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 8623 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ On ∧ 𝑤 ∈ On) → (𝐴 +no 𝑤) ∈ On)
3328, 31, 32syl2anc 584 . . . . . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . . . . 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 6364 . . . . . . . . . . . . . . . . 17 (((𝐴 +no 𝑤) ∈ On ∧ 𝑥 ∈ On) → (((𝐴 +no 𝑤) ⊆ (𝐵 +no 𝑤) ∧ (𝐵 +no 𝑤) ∈ 𝑥) → (𝐴 +no 𝑤) ∈ 𝑥))
3633, 34, 35syl2anc 584 . . . . . . . . . . . . . . . 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 697 . . . . . . . . . . . . . . 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 3143 . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . 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 780 . . . . . . . . . . . . . . 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 4010 . . . . . . . . . . . . . . 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 512 . . . . . . . . . . . . 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 457 . . . . . . . . . . . 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 4033 . . . . . . . . . . 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 4930 . . . . . . . . . . 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 773 . . . . . . . . . . 11 ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → 𝑐 ∈ On)
49 naddov2 8625 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑐 ∈ On) → (𝐴 +no 𝑐) = {𝑥 ∈ On ∣ (∀𝑤𝑐 (𝐴 +no 𝑤) ∈ 𝑥 ∧ ∀𝑧𝐴 (𝑧 +no 𝑐) ∈ 𝑥)})
5026, 48, 49syl2anc 584 . . . . . . . . . 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 776 . . . . . . . . . . . 12 (((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) → 𝐵 ∈ On)
5251adantr 481 . . . . . . . . . . 11 ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → 𝐵 ∈ On)
53 naddov2 8625 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑐 ∈ On) → (𝐵 +no 𝑐) = {𝑥 ∈ On ∣ (∀𝑦𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧𝐵 (𝑧 +no 𝑐) ∈ 𝑥)})
5452, 48, 53syl2anc 584 . . . . . . . . . 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 3991 . . . . . . . . 9 ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴𝐵) ∧ ∀𝑑𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐))
5655exp31 420 . . . . . . . 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 413 . . . . . 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, 59biimtrid 241 . . . 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 7794 . . 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 1117 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 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  {crab 3407  wss 3910   cint 4907  Oncon0 6317  (class class class)co 7357   +no cnadd 8611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-frecs 8212  df-nadd 8612
This theorem is referenced by:  naddel1  8632
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