Theorem onvf1odlem1 35097

Description: Lemma for onvf1od 35101. (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 5274	. . . . . 6 ⊢ ¬ V ∈ 𝑉
2		eleq1 2817	. . . . . . 7 ⊢ (V = 𝐴 → (V ∈ 𝑉 ↔ 𝐴 ∈ 𝑉))
3	2	eqcoms 2738	. . . . . 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 3460	. . . . . 6 ⊢ (𝐴 = V ↔ ∀𝑦 𝑦 ∈ 𝐴)
7		alex 1826	. . . . . 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 1835	. . . 4 ⊢ (∃𝑦 ¬ 𝑦 ∈ 𝐴 → ∃𝑦∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴)
14		rexv 3478	. . . 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 9752	. . . . 5 ⊢ (𝑦 ∈ V → ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥))
17	16	rgen 3047	. . . 4 ⊢ ∀𝑦 ∈ V ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)
18		r19.29r 3097	. . . . 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 3095	. . . . . 6 ⊢ ((∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴 ∧ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)) → ∃𝑥 ∈ On (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)))
20	19	reximi 3068	. . . . 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 3267	. . 3 ⊢ (∃𝑦 ∈ V ∃𝑥 ∈ On (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)) ↔ ∃𝑥 ∈ On ∃𝑦 ∈ V (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)))
25		exancom 1861	. . . . 5 ⊢ (∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)) ↔ ∃𝑦(𝑦 ∈ (𝑅1‘𝑥) ∧ ¬ 𝑦 ∈ 𝐴))
26		rexv 3478	. . . . 5 ⊢ (∃𝑦 ∈ V (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)) ↔ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)))
27		df-rex 3055	. . . . 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 3077	. . 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 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  Vcvv 3450  Oncon0 6335  ‘cfv 6514  𝑅₁cr1 9722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-reg 9552  ax-inf2 9601

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-r1 9724

This theorem is referenced by:  onvf1odlem2  35098  onvf1odlem4  35100