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Mirrors > Home > MPE Home > Th. List > infcllem | Structured version Visualization version GIF version |
Description: Lemma for infcl 8940, inflb 8941, infglb 8942, etc. (Contributed by AV, 3-Sep-2020.) |
Ref | Expression |
---|---|
infcl.1 | ⊢ (𝜑 → 𝑅 Or 𝐴) |
infcl.2 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) |
Ref | Expression |
---|---|
infcllem | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infcl.2 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) | |
2 | vex 3495 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
3 | vex 3495 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | brcnv 5746 | . . . . . . 7 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) |
5 | 4 | bicomi 225 | . . . . . 6 ⊢ (𝑦𝑅𝑥 ↔ 𝑥◡𝑅𝑦) |
6 | 5 | notbii 321 | . . . . 5 ⊢ (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑥◡𝑅𝑦) |
7 | 6 | ralbii 3162 | . . . 4 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦) |
8 | 3, 2 | brcnv 5746 | . . . . . . 7 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦) |
9 | 8 | bicomi 225 | . . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ 𝑦◡𝑅𝑥) |
10 | vex 3495 | . . . . . . . . 9 ⊢ 𝑧 ∈ V | |
11 | 3, 10 | brcnv 5746 | . . . . . . . 8 ⊢ (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦) |
12 | 11 | bicomi 225 | . . . . . . 7 ⊢ (𝑧𝑅𝑦 ↔ 𝑦◡𝑅𝑧) |
13 | 12 | rexbii 3244 | . . . . . 6 ⊢ (∃𝑧 ∈ 𝐵 𝑧𝑅𝑦 ↔ ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧) |
14 | 9, 13 | imbi12i 352 | . . . . 5 ⊢ ((𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) ↔ (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)) |
15 | 14 | ralbii 3162 | . . . 4 ⊢ (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)) |
16 | 7, 15 | anbi12i 626 | . . 3 ⊢ ((∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) ↔ (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))) |
17 | 16 | rexbii 3244 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) ↔ ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))) |
18 | 1, 17 | sylib 219 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∀wral 3135 ∃wrex 3136 class class class wbr 5057 Or wor 5466 ◡ccnv 5547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-cnv 5556 |
This theorem is referenced by: infcl 8940 inflb 8941 infglb 8942 infglbb 8943 infiso 8960 |
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