Theorem onvf1odlem1 35305

Description: Lemma for onvf1od 35309. (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 5254	. . . . . 6 ⊢ ¬ V ∈ 𝑉
2		eleq1 2825	. . . . . . 7 ⊢ (V = 𝐴 → (V ∈ 𝑉 ↔ 𝐴 ∈ 𝑉))
3	2	eqcoms 2745	. . . . . 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 3440	. . . . . 6 ⊢ (𝐴 = V ↔ ∀𝑦 𝑦 ∈ 𝐴)
7		alex 1828	. . . . . 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 3129	. . . . 5 ⊢ (¬ 𝑦 ∈ 𝐴 → ∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴)
13	12	eximi 1837	. . . 4 ⊢ (∃𝑦 ¬ 𝑦 ∈ 𝐴 → ∃𝑦∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴)
14		rexv 3458	. . . 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 9711	. . . . 5 ⊢ (𝑦 ∈ V → ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥))
17	16	rgen 3054	. . . 4 ⊢ ∀𝑦 ∈ V ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)
18		r19.29r 3102	. . . . 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 3101	. . . . . 6 ⊢ ((∀𝑥 ∈ On ¬ 𝑦 ∈ 𝐴 ∧ ∃𝑥 ∈ On 𝑦 ∈ (𝑅1‘𝑥)) → ∃𝑥 ∈ On (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)))
20	19	reximi 3076	. . . . 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 692	. . 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 1863	. . . . 5 ⊢ (∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)) ↔ ∃𝑦(𝑦 ∈ (𝑅1‘𝑥) ∧ ¬ 𝑦 ∈ 𝐴))
26		rexv 3458	. . . . 5 ⊢ (∃𝑦 ∈ V (¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)) ↔ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ (𝑅1‘𝑥)))
27		df-rex 3063	. . . . 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 3085	. . 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 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3430  Oncon0 6319  ‘cfv 6494  𝑅₁cr1 9681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-reg 9502  ax-inf2 9557

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-om 7813  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-r1 9683

This theorem is referenced by:  onvf1odlem2  35306  onvf1odlem4  35308