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Mirrors > Home > MPE Home > Th. List > psseq2 | Structured version Visualization version GIF version |
Description: Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.) |
Ref | Expression |
---|---|
psseq2 | ⊢ (𝐴 = 𝐵 → (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq2 3990 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵)) | |
2 | neeq2 3076 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵)) | |
3 | 1, 2 | anbi12d 630 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐶 ⊆ 𝐴 ∧ 𝐶 ≠ 𝐴) ↔ (𝐶 ⊆ 𝐵 ∧ 𝐶 ≠ 𝐵))) |
4 | df-pss 3951 | . 2 ⊢ (𝐶 ⊊ 𝐴 ↔ (𝐶 ⊆ 𝐴 ∧ 𝐶 ≠ 𝐴)) | |
5 | df-pss 3951 | . 2 ⊢ (𝐶 ⊊ 𝐵 ↔ (𝐶 ⊆ 𝐵 ∧ 𝐶 ≠ 𝐵)) | |
6 | 3, 4, 5 | 3bitr4g 315 | 1 ⊢ (𝐴 = 𝐵 → (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ≠ wne 3013 ⊆ wss 3933 ⊊ wpss 3934 |
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 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-ne 3014 df-in 3940 df-ss 3949 df-pss 3951 |
This theorem is referenced by: psseq2i 4064 psseq2d 4067 psssstr 4080 brrpssg 7440 sorpssint 7448 php 8689 php2 8690 pssnn 8724 isfin4 9707 fin2i2 9728 elnp 10397 elnpi 10398 ltprord 10440 pgpfac1lem1 19125 pgpfac1lem5 19130 lbsextlem4 19862 alexsubALTlem4 22586 spansncv 29357 cvbr 29986 cvcon3 29988 cvnbtwn 29990 cvbr4i 30071 dfon2lem6 32930 dfon2lem7 32931 dfon2lem8 32932 dfon2 32934 lcvbr 36037 lcvnbtwn 36041 lsatcv0 36047 lsat0cv 36049 islshpcv 36069 mapdcv 38676 pssn0 38991 |
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