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Mirrors > Home > MPE Home > Th. List > domeng | Structured version Visualization version GIF version |
Description: Dominance in terms of equinumerosity, with the sethood requirement expressed as an antecedent. Example 1 of [Enderton] p. 146. (Contributed by NM, 24-Apr-2004.) |
Ref | Expression |
---|---|
domeng | ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 5108 | . 2 ⊢ (𝑦 = 𝐵 → (𝐴 ≼ 𝑦 ↔ 𝐴 ≼ 𝐵)) | |
2 | sseq2 3969 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝐵)) | |
3 | 2 | anbi2d 630 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝑦) ↔ (𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵))) |
4 | 3 | exbidv 1925 | . 2 ⊢ (𝑦 = 𝐵 → (∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝑦) ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵))) |
5 | vex 3448 | . . 3 ⊢ 𝑦 ∈ V | |
6 | 5 | domen 8835 | . 2 ⊢ (𝐴 ≼ 𝑦 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝑦)) |
7 | 1, 4, 6 | vtoclbg 3527 | 1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ⊆ wss 3909 class class class wbr 5104 ≈ cen 8814 ≼ cdom 8815 |
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-11 2155 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 ax-un 7663 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-xp 5637 df-rel 5638 df-cnv 5639 df-dm 5641 df-rn 5642 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-en 8818 df-dom 8819 |
This theorem is referenced by: undomOLD 8938 mapdom1 9020 mapdom2 9026 domfi 9070 isfinite2 9179 unxpwdom 9459 djuinf 10058 domfin4 10181 pwfseq 10534 grudomon 10687 ufldom 23241 erdsze2lem1 33577 |
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