Theorem naddle 36801

Description: Condition for bounding natural addition above. (Contributed by Scott Fenton, 21-Jul-2026.)

Assertion

Ref	Expression
naddle	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐵) ⊆ 𝐶 ↔ (∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝐶 ∧ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝐶)))

Distinct variable groups:   𝐴,𝑎,𝑏   𝐵,𝑎,𝑏   𝐶,𝑎,𝑏

Proof of Theorem naddle

Step	Hyp	Ref	Expression
1		ltnadd 36800	. . . 4 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ (𝐴 +no 𝐵) ↔ (∃𝑎 ∈ 𝐴 𝐶 ⊆ (𝑎 +no 𝐵) ∨ ∃𝑏 ∈ 𝐵 𝐶 ⊆ (𝐴 +no 𝑏))))
2	1	3coml 1145	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶 ∈ (𝐴 +no 𝐵) ↔ (∃𝑎 ∈ 𝐴 𝐶 ⊆ (𝑎 +no 𝐵) ∨ ∃𝑏 ∈ 𝐵 𝐶 ⊆ (𝐴 +no 𝑏))))
3	2	notbid 321	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ 𝐶 ∈ (𝐴 +no 𝐵) ↔ ¬ (∃𝑎 ∈ 𝐴 𝐶 ⊆ (𝑎 +no 𝐵) ∨ ∃𝑏 ∈ 𝐵 𝐶 ⊆ (𝐴 +no 𝑏))))
4		naddcl 8668	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) ∈ On)
5	4	3adant3 1150	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 +no 𝐵) ∈ On)
6		simp3 1156	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐶 ∈ On)
7		ontri1 6396	. . 3 ⊢ (((𝐴 +no 𝐵) ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐵) ⊆ 𝐶 ↔ ¬ 𝐶 ∈ (𝐴 +no 𝐵)))
8	5, 6, 7	syl2anc 596	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐵) ⊆ 𝐶 ↔ ¬ 𝐶 ∈ (𝐴 +no 𝐵)))
9		simpl3 1212	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑎 ∈ 𝐴) → 𝐶 ∈ On)
10		onss 7787	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
11	10	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ⊆ On)
12	11	sselda 3934	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ On)
13		simpl2 1211	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑎 ∈ 𝐴) → 𝐵 ∈ On)
14	12, 13	naddcld 8671	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑎 ∈ 𝐴) → (𝑎 +no 𝐵) ∈ On)
15		ontri1 6396	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ (𝑎 +no 𝐵) ∈ On) → (𝐶 ⊆ (𝑎 +no 𝐵) ↔ ¬ (𝑎 +no 𝐵) ∈ 𝐶))
16	9, 14, 15	syl2anc 596	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑎 ∈ 𝐴) → (𝐶 ⊆ (𝑎 +no 𝐵) ↔ ¬ (𝑎 +no 𝐵) ∈ 𝐶))
17	16	rexbidva 3186	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∃𝑎 ∈ 𝐴 𝐶 ⊆ (𝑎 +no 𝐵) ↔ ∃𝑎 ∈ 𝐴 ¬ (𝑎 +no 𝐵) ∈ 𝐶))
18		simpl3 1212	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑏 ∈ 𝐵) → 𝐶 ∈ On)
19		simpl1 1210	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑏 ∈ 𝐵) → 𝐴 ∈ On)
20		onss 7787	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
21	20	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐵 ⊆ On)
22	21	sselda 3934	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ On)
23	19, 22	naddcld 8671	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑏 ∈ 𝐵) → (𝐴 +no 𝑏) ∈ On)
24		ontri1 6396	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ (𝐴 +no 𝑏) ∈ On) → (𝐶 ⊆ (𝐴 +no 𝑏) ↔ ¬ (𝐴 +no 𝑏) ∈ 𝐶))
25	18, 23, 24	syl2anc 596	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑏 ∈ 𝐵) → (𝐶 ⊆ (𝐴 +no 𝑏) ↔ ¬ (𝐴 +no 𝑏) ∈ 𝐶))
26	25	rexbidva 3186	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∃𝑏 ∈ 𝐵 𝐶 ⊆ (𝐴 +no 𝑏) ↔ ∃𝑏 ∈ 𝐵 ¬ (𝐴 +no 𝑏) ∈ 𝐶))
27	17, 26	orbi12d 932	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∃𝑎 ∈ 𝐴 𝐶 ⊆ (𝑎 +no 𝐵) ∨ ∃𝑏 ∈ 𝐵 𝐶 ⊆ (𝐴 +no 𝑏)) ↔ (∃𝑎 ∈ 𝐴 ¬ (𝑎 +no 𝐵) ∈ 𝐶 ∨ ∃𝑏 ∈ 𝐵 ¬ (𝐴 +no 𝑏) ∈ 𝐶)))
28		rexnal 3116	. . . . . 6 ⊢ (∃𝑎 ∈ 𝐴 ¬ (𝑎 +no 𝐵) ∈ 𝐶 ↔ ¬ ∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝐶)
29		rexnal 3116	. . . . . 6 ⊢ (∃𝑏 ∈ 𝐵 ¬ (𝐴 +no 𝑏) ∈ 𝐶 ↔ ¬ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝐶)
30	28, 29	orbi12i 928	. . . . 5 ⊢ ((∃𝑎 ∈ 𝐴 ¬ (𝑎 +no 𝐵) ∈ 𝐶 ∨ ∃𝑏 ∈ 𝐵 ¬ (𝐴 +no 𝑏) ∈ 𝐶) ↔ (¬ ∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝐶 ∨ ¬ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝐶))
31		ianor 997	. . . . 5 ⊢ (¬ (∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝐶 ∧ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝐶) ↔ (¬ ∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝐶 ∨ ¬ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝐶))
32	30, 31	bitr4i 281	. . . 4 ⊢ ((∃𝑎 ∈ 𝐴 ¬ (𝑎 +no 𝐵) ∈ 𝐶 ∨ ∃𝑏 ∈ 𝐵 ¬ (𝐴 +no 𝑏) ∈ 𝐶) ↔ ¬ (∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝐶 ∧ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝐶))
33	27, 32	bitrdi 290	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∃𝑎 ∈ 𝐴 𝐶 ⊆ (𝑎 +no 𝐵) ∨ ∃𝑏 ∈ 𝐵 𝐶 ⊆ (𝐴 +no 𝑏)) ↔ ¬ (∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝐶 ∧ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝐶)))
34	33	con2bid 357	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝐶 ∧ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝐶) ↔ ¬ (∃𝑎 ∈ 𝐴 𝐶 ⊆ (𝑎 +no 𝐵) ∨ ∃𝑏 ∈ 𝐵 𝐶 ⊆ (𝐴 +no 𝑏))))
35	3, 8, 34	3bitr4d 314	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐵) ⊆ 𝐶 ↔ (∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝐶 ∧ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝐶)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088   ⊆ wss 3902  Oncon0 6361  (class class class)co 7416   +no cnadd 8656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7989  df-2nd 7990  df-frecs 8283  df-nadd 8657

This theorem is used by:  nadddilem2  36803