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Theorem fnessref 32673
Description: A cover is finer iff it has a subcover which is both finer and a refinement. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
Hypotheses
Ref Expression
fnessref.1 𝑋 = 𝐴
fnessref.2 𝑌 = 𝐵
Assertion
Ref Expression
fnessref (𝑋 = 𝑌 → (𝐴Fne𝐵 ↔ ∃𝑐(𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))))
Distinct variable groups:   𝐴,𝑐   𝐵,𝑐   𝑋,𝑐   𝑌,𝑐

Proof of Theorem fnessref
Dummy variables 𝑡 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnerel 32654 . . . . . . 7 Rel Fne
21brrelex2i 5359 . . . . . 6 (𝐴Fne𝐵𝐵 ∈ V)
32adantl 469 . . . . 5 ((𝑋 = 𝑌𝐴Fne𝐵) → 𝐵 ∈ V)
4 rabexg 5006 . . . . 5 (𝐵 ∈ V → {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∈ V)
53, 4syl 17 . . . 4 ((𝑋 = 𝑌𝐴Fne𝐵) → {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∈ V)
6 ssrab2 3884 . . . . . 6 {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ⊆ 𝐵
76a1i 11 . . . . 5 ((𝑋 = 𝑌𝐴Fne𝐵) → {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ⊆ 𝐵)
8 fnessref.1 . . . . . . . . . . . 12 𝑋 = 𝐴
98eleq2i 2877 . . . . . . . . . . 11 (𝑡𝑋𝑡 𝐴)
10 eluni 4633 . . . . . . . . . . 11 (𝑡 𝐴 ↔ ∃𝑧(𝑡𝑧𝑧𝐴))
119, 10bitri 266 . . . . . . . . . 10 (𝑡𝑋 ↔ ∃𝑧(𝑡𝑧𝑧𝐴))
12 fnessex 32662 . . . . . . . . . . . . . . . . 17 ((𝐴Fne𝐵𝑧𝐴𝑡𝑧) → ∃𝑥𝐵 (𝑡𝑥𝑥𝑧))
13123expia 1143 . . . . . . . . . . . . . . . 16 ((𝐴Fne𝐵𝑧𝐴) → (𝑡𝑧 → ∃𝑥𝐵 (𝑡𝑥𝑥𝑧)))
1413adantll 696 . . . . . . . . . . . . . . 15 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ 𝑧𝐴) → (𝑡𝑧 → ∃𝑥𝐵 (𝑡𝑥𝑥𝑧)))
15 sseq2 3824 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑧 → (𝑥𝑦𝑥𝑧))
1615rspcev 3502 . . . . . . . . . . . . . . . . . . 19 ((𝑧𝐴𝑥𝑧) → ∃𝑦𝐴 𝑥𝑦)
1716ex 399 . . . . . . . . . . . . . . . . . 18 (𝑧𝐴 → (𝑥𝑧 → ∃𝑦𝐴 𝑥𝑦))
1817adantl 469 . . . . . . . . . . . . . . . . 17 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ 𝑧𝐴) → (𝑥𝑧 → ∃𝑦𝐴 𝑥𝑦))
1918anim2d 601 . . . . . . . . . . . . . . . 16 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ 𝑧𝐴) → ((𝑡𝑥𝑥𝑧) → (𝑡𝑥 ∧ ∃𝑦𝐴 𝑥𝑦)))
2019reximdv 3203 . . . . . . . . . . . . . . 15 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ 𝑧𝐴) → (∃𝑥𝐵 (𝑡𝑥𝑥𝑧) → ∃𝑥𝐵 (𝑡𝑥 ∧ ∃𝑦𝐴 𝑥𝑦)))
2114, 20syld 47 . . . . . . . . . . . . . 14 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ 𝑧𝐴) → (𝑡𝑧 → ∃𝑥𝐵 (𝑡𝑥 ∧ ∃𝑦𝐴 𝑥𝑦)))
2221ex 399 . . . . . . . . . . . . 13 ((𝑋 = 𝑌𝐴Fne𝐵) → (𝑧𝐴 → (𝑡𝑧 → ∃𝑥𝐵 (𝑡𝑥 ∧ ∃𝑦𝐴 𝑥𝑦))))
2322com23 86 . . . . . . . . . . . 12 ((𝑋 = 𝑌𝐴Fne𝐵) → (𝑡𝑧 → (𝑧𝐴 → ∃𝑥𝐵 (𝑡𝑥 ∧ ∃𝑦𝐴 𝑥𝑦))))
2423impd 398 . . . . . . . . . . 11 ((𝑋 = 𝑌𝐴Fne𝐵) → ((𝑡𝑧𝑧𝐴) → ∃𝑥𝐵 (𝑡𝑥 ∧ ∃𝑦𝐴 𝑥𝑦)))
2524exlimdv 2024 . . . . . . . . . 10 ((𝑋 = 𝑌𝐴Fne𝐵) → (∃𝑧(𝑡𝑧𝑧𝐴) → ∃𝑥𝐵 (𝑡𝑥 ∧ ∃𝑦𝐴 𝑥𝑦)))
2611, 25syl5bi 233 . . . . . . . . 9 ((𝑋 = 𝑌𝐴Fne𝐵) → (𝑡𝑋 → ∃𝑥𝐵 (𝑡𝑥 ∧ ∃𝑦𝐴 𝑥𝑦)))
27 elunirab 4642 . . . . . . . . 9 (𝑡 {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ↔ ∃𝑥𝐵 (𝑡𝑥 ∧ ∃𝑦𝐴 𝑥𝑦))
2826, 27syl6ibr 243 . . . . . . . 8 ((𝑋 = 𝑌𝐴Fne𝐵) → (𝑡𝑋𝑡 {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}))
2928ssrdv 3804 . . . . . . 7 ((𝑋 = 𝑌𝐴Fne𝐵) → 𝑋 {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦})
306unissi 4655 . . . . . . . 8 {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ⊆ 𝐵
31 simpl 470 . . . . . . . . 9 ((𝑋 = 𝑌𝐴Fne𝐵) → 𝑋 = 𝑌)
32 fnessref.2 . . . . . . . . 9 𝑌 = 𝐵
3331, 32syl6req 2857 . . . . . . . 8 ((𝑋 = 𝑌𝐴Fne𝐵) → 𝐵 = 𝑋)
3430, 33syl5sseq 3850 . . . . . . 7 ((𝑋 = 𝑌𝐴Fne𝐵) → {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ⊆ 𝑋)
3529, 34eqssd 3815 . . . . . 6 ((𝑋 = 𝑌𝐴Fne𝐵) → 𝑋 = {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦})
36 fnessex 32662 . . . . . . . . . 10 ((𝐴Fne𝐵𝑧𝐴𝑡𝑧) → ∃𝑤𝐵 (𝑡𝑤𝑤𝑧))
37363expb 1142 . . . . . . . . 9 ((𝐴Fne𝐵 ∧ (𝑧𝐴𝑡𝑧)) → ∃𝑤𝐵 (𝑡𝑤𝑤𝑧))
3837adantll 696 . . . . . . . 8 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ (𝑧𝐴𝑡𝑧)) → ∃𝑤𝐵 (𝑡𝑤𝑤𝑧))
39 simpl 470 . . . . . . . . . . . . 13 ((𝑤𝐵 ∧ (𝑡𝑤𝑤𝑧)) → 𝑤𝐵)
4039a1i 11 . . . . . . . . . . . 12 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ (𝑧𝐴𝑡𝑧)) → ((𝑤𝐵 ∧ (𝑡𝑤𝑤𝑧)) → 𝑤𝐵))
41 sseq2 3824 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝑤𝑦𝑤𝑧))
4241rspcev 3502 . . . . . . . . . . . . . . . 16 ((𝑧𝐴𝑤𝑧) → ∃𝑦𝐴 𝑤𝑦)
4342expcom 400 . . . . . . . . . . . . . . 15 (𝑤𝑧 → (𝑧𝐴 → ∃𝑦𝐴 𝑤𝑦))
4443ad2antll 711 . . . . . . . . . . . . . 14 ((𝑤𝐵 ∧ (𝑡𝑤𝑤𝑧)) → (𝑧𝐴 → ∃𝑦𝐴 𝑤𝑦))
4544com12 32 . . . . . . . . . . . . 13 (𝑧𝐴 → ((𝑤𝐵 ∧ (𝑡𝑤𝑤𝑧)) → ∃𝑦𝐴 𝑤𝑦))
4645ad2antrl 710 . . . . . . . . . . . 12 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ (𝑧𝐴𝑡𝑧)) → ((𝑤𝐵 ∧ (𝑡𝑤𝑤𝑧)) → ∃𝑦𝐴 𝑤𝑦))
4740, 46jcad 504 . . . . . . . . . . 11 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ (𝑧𝐴𝑡𝑧)) → ((𝑤𝐵 ∧ (𝑡𝑤𝑤𝑧)) → (𝑤𝐵 ∧ ∃𝑦𝐴 𝑤𝑦)))
48 sseq1 3823 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (𝑥𝑦𝑤𝑦))
4948rexbidv 3240 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (∃𝑦𝐴 𝑥𝑦 ↔ ∃𝑦𝐴 𝑤𝑦))
5049elrab 3559 . . . . . . . . . . 11 (𝑤 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ↔ (𝑤𝐵 ∧ ∃𝑦𝐴 𝑤𝑦))
5147, 50syl6ibr 243 . . . . . . . . . 10 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ (𝑧𝐴𝑡𝑧)) → ((𝑤𝐵 ∧ (𝑡𝑤𝑤𝑧)) → 𝑤 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}))
52 simpr 473 . . . . . . . . . . 11 ((𝑤𝐵 ∧ (𝑡𝑤𝑤𝑧)) → (𝑡𝑤𝑤𝑧))
5352a1i 11 . . . . . . . . . 10 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ (𝑧𝐴𝑡𝑧)) → ((𝑤𝐵 ∧ (𝑡𝑤𝑤𝑧)) → (𝑡𝑤𝑤𝑧)))
5451, 53jcad 504 . . . . . . . . 9 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ (𝑧𝐴𝑡𝑧)) → ((𝑤𝐵 ∧ (𝑡𝑤𝑤𝑧)) → (𝑤 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∧ (𝑡𝑤𝑤𝑧))))
5554reximdv2 3201 . . . . . . . 8 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ (𝑧𝐴𝑡𝑧)) → (∃𝑤𝐵 (𝑡𝑤𝑤𝑧) → ∃𝑤 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} (𝑡𝑤𝑤𝑧)))
5638, 55mpd 15 . . . . . . 7 (((𝑋 = 𝑌𝐴Fne𝐵) ∧ (𝑧𝐴𝑡𝑧)) → ∃𝑤 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} (𝑡𝑤𝑤𝑧))
5756ralrimivva 3159 . . . . . 6 ((𝑋 = 𝑌𝐴Fne𝐵) → ∀𝑧𝐴𝑡𝑧𝑤 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} (𝑡𝑤𝑤𝑧))
58 eqid 2806 . . . . . . . 8 {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} = {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}
598, 58isfne2 32658 . . . . . . 7 ({𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∈ V → (𝐴Fne{𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ↔ (𝑋 = {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∧ ∀𝑧𝐴𝑡𝑧𝑤 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} (𝑡𝑤𝑤𝑧))))
603, 4, 593syl 18 . . . . . 6 ((𝑋 = 𝑌𝐴Fne𝐵) → (𝐴Fne{𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ↔ (𝑋 = {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∧ ∀𝑧𝐴𝑡𝑧𝑤 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} (𝑡𝑤𝑤𝑧))))
6135, 57, 60mpbir2and 695 . . . . 5 ((𝑋 = 𝑌𝐴Fne𝐵) → 𝐴Fne{𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦})
62 sseq1 3823 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥𝑦𝑧𝑦))
6362rexbidv 3240 . . . . . . . . 9 (𝑥 = 𝑧 → (∃𝑦𝐴 𝑥𝑦 ↔ ∃𝑦𝐴 𝑧𝑦))
6463elrab 3559 . . . . . . . 8 (𝑧 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ↔ (𝑧𝐵 ∧ ∃𝑦𝐴 𝑧𝑦))
65 sseq2 3824 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝑧𝑦𝑧𝑤))
6665cbvrexv 3361 . . . . . . . . . . 11 (∃𝑦𝐴 𝑧𝑦 ↔ ∃𝑤𝐴 𝑧𝑤)
6766biimpi 207 . . . . . . . . . 10 (∃𝑦𝐴 𝑧𝑦 → ∃𝑤𝐴 𝑧𝑤)
6867adantl 469 . . . . . . . . 9 ((𝑧𝐵 ∧ ∃𝑦𝐴 𝑧𝑦) → ∃𝑤𝐴 𝑧𝑤)
6968a1i 11 . . . . . . . 8 ((𝑋 = 𝑌𝐴Fne𝐵) → ((𝑧𝐵 ∧ ∃𝑦𝐴 𝑧𝑦) → ∃𝑤𝐴 𝑧𝑤))
7064, 69syl5bi 233 . . . . . . 7 ((𝑋 = 𝑌𝐴Fne𝐵) → (𝑧 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} → ∃𝑤𝐴 𝑧𝑤))
7170ralrimiv 3153 . . . . . 6 ((𝑋 = 𝑌𝐴Fne𝐵) → ∀𝑧 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}∃𝑤𝐴 𝑧𝑤)
7258, 8isref 21526 . . . . . . 7 ({𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∈ V → ({𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}Ref𝐴 ↔ (𝑋 = {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∧ ∀𝑧 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}∃𝑤𝐴 𝑧𝑤)))
733, 4, 723syl 18 . . . . . 6 ((𝑋 = 𝑌𝐴Fne𝐵) → ({𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}Ref𝐴 ↔ (𝑋 = {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∧ ∀𝑧 ∈ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}∃𝑤𝐴 𝑧𝑤)))
7435, 71, 73mpbir2and 695 . . . . 5 ((𝑋 = 𝑌𝐴Fne𝐵) → {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}Ref𝐴)
757, 61, 74jca32 507 . . . 4 ((𝑋 = 𝑌𝐴Fne𝐵) → ({𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ⊆ 𝐵 ∧ (𝐴Fne{𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∧ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}Ref𝐴)))
76 sseq1 3823 . . . . . 6 (𝑐 = {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} → (𝑐𝐵 ↔ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ⊆ 𝐵))
77 breq2 4848 . . . . . . 7 (𝑐 = {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} → (𝐴Fne𝑐𝐴Fne{𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}))
78 breq1 4847 . . . . . . 7 (𝑐 = {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} → (𝑐Ref𝐴 ↔ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}Ref𝐴))
7977, 78anbi12d 618 . . . . . 6 (𝑐 = {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} → ((𝐴Fne𝑐𝑐Ref𝐴) ↔ (𝐴Fne{𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∧ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}Ref𝐴)))
8076, 79anbi12d 618 . . . . 5 (𝑐 = {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} → ((𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴)) ↔ ({𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ⊆ 𝐵 ∧ (𝐴Fne{𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∧ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}Ref𝐴))))
8180spcegv 3487 . . . 4 ({𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∈ V → (({𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ⊆ 𝐵 ∧ (𝐴Fne{𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦} ∧ {𝑥𝐵 ∣ ∃𝑦𝐴 𝑥𝑦}Ref𝐴)) → ∃𝑐(𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))))
825, 75, 81sylc 65 . . 3 ((𝑋 = 𝑌𝐴Fne𝐵) → ∃𝑐(𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴)))
8382ex 399 . 2 (𝑋 = 𝑌 → (𝐴Fne𝐵 → ∃𝑐(𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))))
84 simprrl 790 . . . . 5 ((𝑋 = 𝑌 ∧ (𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝐴Fne𝑐)
85 eqid 2806 . . . . . . . . . . . 12 𝑐 = 𝑐
868, 85fnebas 32660 . . . . . . . . . . 11 (𝐴Fne𝑐𝑋 = 𝑐)
8784, 86syl 17 . . . . . . . . . 10 ((𝑋 = 𝑌 ∧ (𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑋 = 𝑐)
88 simpl 470 . . . . . . . . . 10 ((𝑋 = 𝑌 ∧ (𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑋 = 𝑌)
8987, 88eqtr3d 2842 . . . . . . . . 9 ((𝑋 = 𝑌 ∧ (𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑐 = 𝑌)
9089, 32syl6eq 2856 . . . . . . . 8 ((𝑋 = 𝑌 ∧ (𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑐 = 𝐵)
91 vuniex 7184 . . . . . . . 8 𝑐 ∈ V
9290, 91syl6eqelr 2894 . . . . . . 7 ((𝑋 = 𝑌 ∧ (𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝐵 ∈ V)
93 uniexb 7203 . . . . . . 7 (𝐵 ∈ V ↔ 𝐵 ∈ V)
9492, 93sylibr 225 . . . . . 6 ((𝑋 = 𝑌 ∧ (𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝐵 ∈ V)
95 simprl 778 . . . . . 6 ((𝑋 = 𝑌 ∧ (𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑐𝐵)
9685, 32fness 32665 . . . . . 6 ((𝐵 ∈ V ∧ 𝑐𝐵 𝑐 = 𝑌) → 𝑐Fne𝐵)
9794, 95, 89, 96syl3anc 1483 . . . . 5 ((𝑋 = 𝑌 ∧ (𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝑐Fne𝐵)
98 fnetr 32667 . . . . 5 ((𝐴Fne𝑐𝑐Fne𝐵) → 𝐴Fne𝐵)
9984, 97, 98syl2anc 575 . . . 4 ((𝑋 = 𝑌 ∧ (𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))) → 𝐴Fne𝐵)
10099ex 399 . . 3 (𝑋 = 𝑌 → ((𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴)) → 𝐴Fne𝐵))
101100exlimdv 2024 . 2 (𝑋 = 𝑌 → (∃𝑐(𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴)) → 𝐴Fne𝐵))
10283, 101impbid 203 1 (𝑋 = 𝑌 → (𝐴Fne𝐵 ↔ ∃𝑐(𝑐𝐵 ∧ (𝐴Fne𝑐𝑐Ref𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wex 1859  wcel 2156  wral 3096  wrex 3097  {crab 3100  Vcvv 3391  wss 3769   cuni 4630   class class class wbr 4844  Refcref 21519  Fnecfne 32652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-iota 6064  df-fun 6103  df-fv 6109  df-topgen 16309  df-ref 21522  df-fne 32653
This theorem is referenced by: (None)
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