Theorem naddle 37043

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 37042	. . . 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 8687	. . . 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 6397	. . 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 7799	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
11	10	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ⊆ On)
12	11	sselda 3908	. . . . . . . 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 8690	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑎 ∈ 𝐴) → (𝑎 +no 𝐵) ∈ On)
15		ontri1 6397	. . . . . . 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 3184	. . . . 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 7799	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
21	20	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐵 ⊆ On)
22	21	sselda 3908	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ On)
23	19, 22	naddcld 8690	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑏 ∈ 𝐵) → (𝐴 +no 𝑏) ∈ On)
24		ontri1 6397	. . . . . . 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 3184	. . . . 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 3114	. . . . . 6 ⊢ (∃𝑎 ∈ 𝐴 ¬ (𝑎 +no 𝐵) ∈ 𝐶 ↔ ¬ ∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝐶)
29		rexnal 3114	. . . . . 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 2144  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895  Oncon0 6362  (class class class)co 7420   +no cnadd 8675

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-frecs 8300  df-nadd 8676

This theorem is used by:  nadddilem2  37045