Theorem onvf1odlem1 35246

Description: Lemma for onvf1od 35250. (Contributed by BTernaryTau, 2-Dec-2025.)

Assertion

Ref	Expression
onvf1odlem1	⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴)

Distinct variable group:   𝑥,𝐴,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem onvf1odlem1

Step	Hyp	Ref	Expression
1		nvel 5259	. . . . . 6 ⊢ ¬ V ∈ 𝑉
2		eleq1 2822	. . . . . . 7 ⊢ (V = 𝐴 → (V ∈ 𝑉 ↔ 𝐴 ∈ 𝑉))
3	2	eqcoms 2742	. . . . . 6 ⊢ (𝐴 = V → (V ∈ 𝑉 ↔ 𝐴 ∈ 𝑉))
4	1, 3	mtbii 326	. . . . 5 ⊢ (𝐴 = V → ¬ 𝐴 ∈ 𝑉)
5	4	con2i 139	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝐴 = V)
6		eqv 3448	. . . . . 6 ⊢ (𝐴 = V ↔ ∀𝑦 𝑦 ∈ 𝐴)
7		alex 1827	. . . . . 6 ⊢ (∀𝑦 𝑦 ∈ 𝐴 ↔ ¬ ∃𝑦 ¬ 𝑦 ∈ 𝐴)
8	6, 7	bitri 275	. . . . 5 ⊢ (𝐴 = V ↔ ¬ ∃𝑦 ¬ 𝑦 ∈ 𝐴)
9	8	con2bii 357	. . . 4 ⊢ (∃𝑦 ¬ 𝑦 ∈ 𝐴 ↔ ¬ 𝐴 = V)
10	5, 9	sylibr 234	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦 ¬ 𝑦 ∈ 𝐴)
11		ax-1 6	. . . . . 6 ⊢ (¬ 𝑦 ∈ 𝐴 → (𝑥 ∈ On → ¬ 𝑦 ∈ 𝐴))
12	11	ralrimiv 3125	. . . . 5 ⊢ (¬ 𝑦 ∈ 𝐴 → ∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴)
13	12	eximi 1836	. . . 4 ⊢ (∃𝑦 ¬ 𝑦 ∈ 𝐴 → ∃𝑦∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴)
14		rexv 3466	. . . 4 ⊢ (∃𝑦 ∈ V ∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴 ↔ ∃𝑦∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴)
15	13, 14	sylibr 234	. . 3 ⊢ (∃𝑦 ¬ 𝑦 ∈ 𝐴 → ∃𝑦 ∈ V ∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴)
16		tz9.13g 9702	. . . . 5 ⊢ (𝑦 ∈ V → ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥))
17	16	rgen 3051	. . . 4 ⊢ ∀𝑦 ∈ V ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)
18		r19.29r 3098	. . . . 5 ⊢ ((∃𝑦 ∈ V ∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴 ∧ ∀𝑦 ∈ V ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)) → ∃𝑦 ∈ V (∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴 ∧ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)))
19		r19.29 3097	. . . . . 6 ⊢ ((∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴 ∧ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)) → ∃𝑥 ∈ On (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)))
20	19	reximi 3072	. . . . 5 ⊢ (∃𝑦 ∈ V (∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴 ∧ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)) → ∃𝑦 ∈ V ∃𝑥 ∈ On (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)))
21	18, 20	syl 17	. . . 4 ⊢ ((∃𝑦 ∈ V ∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴 ∧ ∀𝑦 ∈ V ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)) → ∃𝑦 ∈ V ∃𝑥 ∈ On (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)))
22	17, 21	mpan2 691	. . 3 ⊢ (∃𝑦 ∈ V ∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴 → ∃𝑦 ∈ V ∃𝑥 ∈ On (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)))
23	10, 15, 22	3syl 18	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦 ∈ V ∃𝑥 ∈ On (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)))
24		rexcom 3263	. . 3 ⊢ (∃𝑦 ∈ V ∃𝑥 ∈ On (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)) ↔ ∃𝑥 ∈ On ∃𝑦 ∈ V (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)))
25		exancom 1862	. . . . 5 ⊢ (∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)) ↔ ∃𝑦(𝑦 ∈ (𝑅1‘𝑥) ∧ ¬ 𝑦 ∈ 𝐴))
26		rexv 3466	. . . . 5 ⊢ (∃𝑦 ∈ V (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)) ↔ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)))
27		df-rex 3059	. . . . 5 ⊢ (∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴 ↔ ∃𝑦(𝑦 ∈ (𝑅1‘𝑥) ∧ ¬ 𝑦 ∈ 𝐴))
28	25, 26, 27	3bitr4i 303	. . . 4 ⊢ (∃𝑦 ∈ V (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)) ↔ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴)
29	28	rexbii 3081	. . 3 ⊢ (∃𝑥 ∈ On ∃𝑦 ∈ V (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)) ↔ ∃𝑥 ∈ On ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴)
30	24, 29	bitri 275	. 2 ⊢ (∃𝑦 ∈ V ∃𝑥 ∈ On (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)) ↔ ∃𝑥 ∈ On ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴)
31	23, 30	sylib 218	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058  Vcvv 3438  Oncon0 6315  ‘cfv 6490  𝑅₁cr1 9672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-reg 9495  ax-inf2 9548

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-r1 9674

This theorem is referenced by:  onvf1odlem2  35247  onvf1odlem4  35249