Theorem onvf1odlem1 35649

Description: Lemma for onvf1od 35653. (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 5284	. . . . . 6 ⊢ ¬ V ∈ 𝑉
2		eleq1 2853	. . . . . . 7 ⊢ (V = 𝐴 → (V ∈ 𝑉 ↔ 𝐴 ∈ 𝑉))
3	2	eqcoms 2773	. . . . . 6 ⊢ (𝐴 = V → (V ∈ 𝑉 ↔ 𝐴 ∈ 𝑉))
4	1, 3	mtbii 329	. . . . 5 ⊢ (𝐴 = V → ¬ 𝐴 ∈ 𝑉)
5	4	con2i 140	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝐴 = V)
6		eqv 3467	. . . . . 6 ⊢ (𝐴 = V ↔ ∀𝑦 𝑦 ∈ 𝐴)
7		alex 1859	. . . . . 6 ⊢ (∀𝑦 𝑦 ∈ 𝐴 ↔ ¬ ∃𝑦 ¬ 𝑦 ∈ 𝐴)
8	6, 7	bitri 278	. . . . 5 ⊢ (𝐴 = V ↔ ¬ ∃𝑦 ¬ 𝑦 ∈ 𝐴)
9	8	con2bii 360	. . . 4 ⊢ (∃𝑦 ¬ 𝑦 ∈ 𝐴 ↔ ¬ 𝐴 = V)
10	5, 9	sylibr 237	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦 ¬ 𝑦 ∈ 𝐴)
11		ax-1 6	. . . . . 6 ⊢ (¬ 𝑦 ∈ 𝐴 → (𝑥 ∈ On → ¬ 𝑦 ∈ 𝐴))
12	11	ralrimiv 3158	. . . . 5 ⊢ (¬ 𝑦 ∈ 𝐴 → ∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴)
13	12	eximi 1868	. . . 4 ⊢ (∃𝑦 ¬ 𝑦 ∈ 𝐴 → ∃𝑦∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴)
14		rexv 3484	. . . 4 ⊢ (∃𝑦 ∈ V ∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴 ↔ ∃𝑦∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴)
15	13, 14	sylibr 237	. . 3 ⊢ (∃𝑦 ¬ 𝑦 ∈ 𝐴 → ∃𝑦 ∈ V ∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴)
16		tz9.13g 9772	. . . . 5 ⊢ (𝑦 ∈ V → ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥))
17	16	rgen 3083	. . . 4 ⊢ ∀𝑦 ∈ V ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)
18		r19.29r 3131	. . . . 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 3130	. . . . . 6 ⊢ ((∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴 ∧ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)) → ∃𝑥 ∈ On (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)))
20	19	reximi 3105	. . . . 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 18	. . . 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 704	. . 3 ⊢ (∃𝑦 ∈ V ∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴 → ∃𝑦 ∈ V ∃𝑥 ∈ On (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)))
23	10, 15, 22	3syl 19	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦 ∈ V ∃𝑥 ∈ On (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)))
24		rexcom 3296	. . 3 ⊢ (∃𝑦 ∈ V ∃𝑥 ∈ On (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)) ↔ ∃𝑥 ∈ On ∃𝑦 ∈ V (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)))
25		exancom 1894	. . . . 5 ⊢ (∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)) ↔ ∃𝑦(𝑦 ∈ (𝑅1‘𝑥) ∧ ¬ 𝑦 ∈ 𝐴))
26		rexv 3484	. . . . 5 ⊢ (∃𝑦 ∈ V (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)) ↔ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)))
27		df-rex 3092	. . . . 5 ⊢ (∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴 ↔ ∃𝑦(𝑦 ∈ (𝑅1‘𝑥) ∧ ¬ 𝑦 ∈ 𝐴))
28	25, 26, 27	3bitr4i 306	. . . 4 ⊢ (∃𝑦 ∈ V (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)) ↔ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴)
29	28	rexbii 3114	. . 3 ⊢ (∃𝑥 ∈ On ∃𝑦 ∈ V (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)) ↔ ∃𝑥 ∈ On ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴)
30	24, 29	bitri 278	. 2 ⊢ (∃𝑦 ∈ V ∃𝑥 ∈ On (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)) ↔ ∃𝑥 ∈ On ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴)
31	23, 30	sylib 221	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  Vcvv 3457  Oncon0 6365  ‘cfv 6541  𝑅₁cr1 9742

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7743  ax-reg 9562  ax-inf2 9618

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7423  df-om 7870  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-r1 9744

This theorem is used by:  onvf1odlem2  35650  onvf1odlem4  35652