Theorem naddle 36719

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 36718	. . . 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 8659	. . . 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 6395	. . 3 ⊢ (((𝐴 +no 𝐵) ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐵) ⊆ 𝐶 ↔ ¬ 𝐶 ∈ (𝐴 +no 𝐵)))
8	5, 6, 7	syl2anc 595	. 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 7780	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
11	10	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ⊆ On)
12	11	sselda 3937	. . . . . . . 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 8662	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑎 ∈ 𝐴) → (𝑎 +no 𝐵) ∈ On)
15		ontri1 6395	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ (𝑎 +no 𝐵) ∈ On) → (𝐶 ⊆ (𝑎 +no 𝐵) ↔ ¬ (𝑎 +no 𝐵) ∈ 𝐶))
16	9, 14, 15	syl2anc 595	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑎 ∈ 𝐴) → (𝐶 ⊆ (𝑎 +no 𝐵) ↔ ¬ (𝑎 +no 𝐵) ∈ 𝐶))
17	16	rexbidva 3187	. . . . 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 7780	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
21	20	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐵 ⊆ On)
22	21	sselda 3937	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ On)
23	19, 22	naddcld 8662	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑏 ∈ 𝐵) → (𝐴 +no 𝑏) ∈ On)
24		ontri1 6395	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ (𝐴 +no 𝑏) ∈ On) → (𝐶 ⊆ (𝐴 +no 𝑏) ↔ ¬ (𝐴 +no 𝑏) ∈ 𝐶))
25	18, 23, 24	syl2anc 595	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑏 ∈ 𝐵) → (𝐶 ⊆ (𝐴 +no 𝑏) ↔ ¬ (𝐴 +no 𝑏) ∈ 𝐶))
26	25	rexbidva 3187	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∃𝑏 ∈ 𝐵 𝐶 ⊆ (𝐴 +no 𝑏) ↔ ∃𝑏 ∈ 𝐵 ¬ (𝐴 +no 𝑏) ∈ 𝐶))
27	17, 26	orbi12d 931	. . . 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 3117	. . . . . 6 ⊢ (∃𝑎 ∈ 𝐴 ¬ (𝑎 +no 𝐵) ∈ 𝐶 ↔ ¬ ∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝐶)
29		rexnal 3117	. . . . . 6 ⊢ (∃𝑏 ∈ 𝐵 ¬ (𝐴 +no 𝑏) ∈ 𝐶 ↔ ¬ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝐶)
30	28, 29	orbi12i 927	. . . . 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 400   ∨ wo 860   ∧ w3a 1103   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ⊆ wss 3905  Oncon0 6360  (class class class)co 7410   +no cnadd 8647

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-frecs 8274  df-nadd 8648

This theorem is used by:  nadddilem2  36721