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Mirrors > Home > MPE Home > Th. List > tz6.26OLD | Structured version Visualization version GIF version |
Description: Obsolete proof of tz6.26 6349 as of 17-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) |
Ref | Expression |
---|---|
tz6.26OLD | ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wereu2 5674 | . . 3 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃!𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑦) | |
2 | reurex 3381 | . . 3 ⊢ (∃!𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑦 → ∃𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑦) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑦) |
4 | rabeq0 4385 | . . . 4 ⊢ ({𝑥 ∈ 𝐵 ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑦) | |
5 | dfrab3 4310 | . . . . . 6 ⊢ {𝑥 ∈ 𝐵 ∣ 𝑥𝑅𝑦} = (𝐵 ∩ {𝑥 ∣ 𝑥𝑅𝑦}) | |
6 | vex 3479 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
7 | 6 | dfpred2 6311 | . . . . . 6 ⊢ Pred(𝑅, 𝐵, 𝑦) = (𝐵 ∩ {𝑥 ∣ 𝑥𝑅𝑦}) |
8 | 5, 7 | eqtr4i 2764 | . . . . 5 ⊢ {𝑥 ∈ 𝐵 ∣ 𝑥𝑅𝑦} = Pred(𝑅, 𝐵, 𝑦) |
9 | 8 | eqeq1i 2738 | . . . 4 ⊢ ({𝑥 ∈ 𝐵 ∣ 𝑥𝑅𝑦} = ∅ ↔ Pred(𝑅, 𝐵, 𝑦) = ∅) |
10 | 4, 9 | bitr3i 277 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ↔ Pred(𝑅, 𝐵, 𝑦) = ∅) |
11 | 10 | rexbii 3095 | . 2 ⊢ (∃𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ↔ ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅) |
12 | 3, 11 | sylib 217 | 1 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 {cab 2710 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 ∃!wreu 3375 {crab 3433 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 class class class wbr 5149 Se wse 5630 We wwe 5631 Predcpred 6300 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-cnv 5685 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 |
This theorem is referenced by: (None) |
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