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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 4790 | . 2 ⊢ (𝑦 = 𝐵 → (𝐴 ≼ 𝑦 ↔ 𝐴 ≼ 𝐵)) | |
2 | sseq2 3776 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝐵)) | |
3 | 2 | anbi2d 614 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝑦) ↔ (𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵))) |
4 | 3 | exbidv 2002 | . 2 ⊢ (𝑦 = 𝐵 → (∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝑦) ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵))) |
5 | vex 3354 | . . 3 ⊢ 𝑦 ∈ V | |
6 | 5 | domen 8120 | . 2 ⊢ (𝐴 ≼ 𝑦 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝑦)) |
7 | 1, 4, 6 | vtoclbg 3418 | 1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∃wex 1852 ∈ wcel 2145 ⊆ wss 3723 class class class wbr 4786 ≈ cen 8104 ≼ cdom 8105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 ax-un 7094 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-xp 5255 df-rel 5256 df-cnv 5257 df-dm 5259 df-rn 5260 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-en 8108 df-dom 8109 |
This theorem is referenced by: undom 8202 mapdom1 8279 mapdom2 8285 domfi 8335 isfinite2 8372 unxpwdom 8648 domfin4 9333 pwfseq 9686 grudomon 9839 ufldom 21979 erdsze2lem1 31516 |
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